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<title>Rinderproblem des Archimedes</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Rinderproblem des Archimedes</h1>

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Das Rinderproblem des Archimedes, auch Problema Bovinum, ist die abgeschwächte Version eines unlösbaren<a href="#cite_note-1993_Schreiber-1">[1]</a> zahlentheoretischen Problems aus der Theorie <a href="Diophantische_Gleichung" title="Diophantische Gleichung">diophantischer Gleichungen</a>, das heißt von <a href="Polynom" title="Polynom">Polynomgleichungen</a> über den <a href="Ganze_Zahl" title="Ganze Zahl">ganzen Zahlen</a>. Das ursprüngliche Problem wird <a href="Archimedes" title="Archimedes">Archimedes</a> zugeschrieben: Die Anzahl der Rinder (Bullen und Kühe, mit je vier Sorten) in einer Herde des Sonnengottes soll bestimmt werden aus einigen Nebenbedingungen.

<div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2></div>

Das Rinderproblem wurde 1773 von <a href="Gotthold_Ephraim_Lessing" title="Gotthold Ephraim Lessing">Gotthold Ephraim Lessing</a> in einem griechischen Manuskript der <a href="Herzog_August_Bibliothek" title="Herzog August Bibliothek">Herzog August Bibliothek</a> in <a href="Wolfenb%C3%BCttel" title="Wolfenbüttel">Wolfenbüttel</a> entdeckt (Cod. Guelf. 77 Gud. Graec.), das einen in 44 <a href="Distichon" title="Distichon">Distichen</a> abgefassten Brief des Archimedes an <a href="Eratosthenes_von_Kyrene" class="mw-redirect" title="Eratosthenes von Kyrene">Eratosthenes von Kyrene</a> enthielt (also aus <a href="Syrakus" title="Syrakus">Syrakus</a> nach <a href="Alexandria" title="Alexandria">Alexandria</a>).<a href="#cite_note-2">[2]</a>

Πληθὺν Ἠελίοιο βοῶν, ὦ ξεῖνε, μέτρησον φροντίδ' ἐπιστήσας, εἰ μετέχεις σοφίης, πόσση ἄρ' ἐν πεδίοις Σικελῆς ποτ' ἐβόσκετο νήσου Θρινακίης τετραχῇ στίφεα δασσαμένη χροιὴν ἀλάσσοντα· τὸ μὲν λευκοῖο γάλακτος, κυανέῳ δ' ἕτερον χρώματι λαμπόμενον, ἄλλο γε μὲν ξανθόν, τὸ δὲ ποικίλον...<a href="#cite_note-3">[3]</a> (Die Menge von Helio's Herde, mein Freund, bemiss ...) – es geht streng übersetzt nicht um Rinder.

Ob der Brief tatsächlich von Archimedes stammt, wird von Lessing und anderen angezweifelt,<a href="#cite_note-4">[4]</a> das Problem selbst ist aufgrund seiner Schwierigkeit jedoch möglicherweise auf Archimedes zurückzuführen.<a href="#cite_note-:0-5">[5]</a> Ein Hinweis darauf ist auch Archimedes’ Interesse an großen Zahlen, wie sie etwa in Der Sandrechner zum Vorschein kommt.

Eine philologische Version des griechischen Textes und eine Übersetzung ins Lateinische findet sich im zweiten Band der von <a href="Johan_Ludvig_Heiberg_(Philologe)" title="Johan Ludvig Heiberg (Philologe)">Johan Ludvig Heiberg</a> besorgten Ausgabe der Werke von Archimedes.<a href="#cite_note-6">[6]</a> Eine deutsche Übertragung des Gedichts wurde von <a href="Georg_Nesselmann" class="mw-redirect" title="Georg Nesselmann">Georg Nesselmann</a> angefertigt und veröffentlicht (1842),<a href="#cite_note-7">[7]</a> eine weitere von Bernhard Krumbiegel (1880).

Der von Lessing veröffentlichte Text enthält eine Teillösung, die aber eine Zeile des Urtextes außer Acht lässt und zwei Forderungen aus dem zweiten Teil des Gedichtes nicht erfüllt. Auch die abgeschwächte Version ohne diese Zusatzforderungen blieb wegen der zur Lösung nötigen Berechnung von sehr großen Zahlen bis vor einigen Jahren ungelöst. Ein Lösungsverfahren wurde 1880 von <a href="August_Amthor" title="August Amthor">August Amthor</a> gefunden, mit dem die Lösung von etwa 7,76 ‧ 10206544 Rindern (eine Zahl mit 206.545 Stellen) bestimmt werden konnte.<a href="#cite_note-:0-5">[5]</a> Für die Berechnung der expliziten Dezimaldarstellung brauchten die Computer (IBM 7040 und <a href="IBM_1620" title="IBM 1620">IBM 1620</a>) von <a href="Hugh_C._Williams" title="Hugh C. Williams">Hugh C. Williams</a>, Gus German und Bob Zarnke 1965 eine Gesamtrechenzeit von 7 Stunden 49 Minuten.<a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Problem">Problem</h2></div>

Das Problem, in einer an Nesselmann und Krumbiegel angelehnten, das Versmaß nicht erhaltenden vereinfachten Fassung:

<dl><dd>Zähle, mein Freund, die Rinder unter der Sonne, die einst unter der Sonne Siziliens grasten, die nach ihrer Farbe in vier Herden geteilt werden. Eine ist milchweiß, eine schwarz, eine gefleckt und eine gelb. Die Anzahl der Bullen jeder Farbe ist größer als die der Kühe dieser Farbe (dies wird in der modernen Version fortgelassen),<a href="#cite_note-1993_Schreiber-1">[1]</a> und die Beziehung zwischen ihnen ist wie folgt:</dd></dl>

<td>weiße Bullen</td>

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<annotation encoding="application/x-tex">{\displaystyle =\left({\tfrac {1}{2}}+{\tfrac {1}{3}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a44373793aed072b2f381d26b43aa5361824c2a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.385ex; height:4.843ex;" alt="{\displaystyle =\left({\tfrac {1}{2}}+{\tfrac {1}{3}}\right)}" loading="lazy"></td>

<td>schwarze Bullen + gelbe Bullen,

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<td>schwarze Bullen</td>

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<annotation encoding="application/x-tex">{\displaystyle =\left({\tfrac {1}{4}}+{\tfrac {1}{5}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8721618ca550fa55297471bb8a6ea15a0cc5d800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.385ex; height:4.843ex;" alt="{\displaystyle =\left({\tfrac {1}{4}}+{\tfrac {1}{5}}\right)}" loading="lazy"></td>

<td>gefleckte Bullen + gelbe Bullen,

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<td>gefleckte Bullen</td>

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<annotation encoding="application/x-tex">{\displaystyle =\left({\tfrac {1}{6}}+{\tfrac {1}{7}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/271fe6682a6ba533a78d382176aa65143f097063.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.385ex; height:4.843ex;" alt="{\displaystyle =\left({\tfrac {1}{6}}+{\tfrac {1}{7}}\right)}" loading="lazy"></td>

<td>weiße Bullen + gelbe Bullen,

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<tr>

<td>weiße Kühe</td>

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<annotation encoding="application/x-tex">{\displaystyle =\left({\tfrac {1}{3}}+{\tfrac {1}{4}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b39a6b95e24d25db9d8a2ceee14900d8c18bbbef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.385ex; height:4.843ex;" alt="{\displaystyle =\left({\tfrac {1}{3}}+{\tfrac {1}{4}}\right)}" loading="lazy"></td>

<td>schwarze Herde,

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<td>schwarze Kühe</td>

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<annotation encoding="application/x-tex">{\displaystyle =\left({\tfrac {1}{4}}+{\tfrac {1}{5}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8721618ca550fa55297471bb8a6ea15a0cc5d800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.385ex; height:4.843ex;" alt="{\displaystyle =\left({\tfrac {1}{4}}+{\tfrac {1}{5}}\right)}" loading="lazy"></td>

<td>gefleckte Herde,

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<td>gefleckte Kühe</td>

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<annotation encoding="application/x-tex">{\displaystyle =\left({\tfrac {1}{5}}+{\tfrac {1}{6}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/179255784f517b4c4c2350c4050fa2bbf53d45ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.385ex; height:4.843ex;" alt="{\displaystyle =\left({\tfrac {1}{5}}+{\tfrac {1}{6}}\right)}" loading="lazy"></td>

<td>gelbe Herde,

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<td>gelbe Kühe</td>

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<annotation encoding="application/x-tex">{\displaystyle =\left({\tfrac {1}{6}}+{\tfrac {1}{7}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/271fe6682a6ba533a78d382176aa65143f097063.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.385ex; height:4.843ex;" alt="{\displaystyle =\left({\tfrac {1}{6}}+{\tfrac {1}{7}}\right)}" loading="lazy"></td>

<td>weiße Herde.

</td></tr></tbody></table></dd></dl></dd></dl>

<dl><dd>Falls du, o Freund, mir nicht die Anzahl der Rinder jeder Art, Bullen und Kühe, angeben kannst, kannst du dich noch nicht als hoch qualifiziert betrachten. Bedenke aber noch die folgenden zusätzlichen Beziehungen zwischen den Bullen unter der Sonne:</dd></dl>

<td>Weiße Bullen + schwarze Bullen</td>

<td>= eine quadratische Zahl,

</td></tr>

<tr>

<td>Gefleckte Bullen + gelbe Bullen</td>

<td>= eine <a href="Dreieckszahl" title="Dreieckszahl">Dreieckszahl</a>.

</td></tr></tbody></table></dd></dl></dd></dl>

<dl><dd>Wenn du diese auch noch berechnet hast, o Freund, und du die Gesamtzahl der Rinder gefunden hast, dann juble als ein Eroberer, weil du dir selbst bewiesen hast, dass du ein sehr begabter Rechner bist.<a href="#cite_note-9">[9]</a></dd></dl>

In Gleichungsform formuliert: Gesucht werden die Anzahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W,X,Y,Z}">

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<annotation encoding="application/x-tex">{\displaystyle W,X,Y,Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b41e96e8b67d6ba14da7b8450065635e4a1961fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.971ex; height:2.509ex;" alt="{\displaystyle W,X,Y,Z}" loading="lazy"> verschieden gefärbter Bullen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w,x,y,z}">

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<annotation encoding="application/x-tex">{\displaystyle w,x,y,z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0f4eccb8967a0d5b5c01afd77c7cfc33a61330f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.339ex; height:2.009ex;" alt="{\displaystyle w,x,y,z}" loading="lazy"> von Kühen in den entsprechenden Farben mit:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W&=\;{\tfrac {5}{6}}X+Z\\X&={\tfrac {9}{20}}Y+Z\\Y&={\tfrac {13}{42}}W+Z\\w&={\tfrac {7}{12}}\,(X+x)\\x&={\tfrac {9}{20}}\,(Y+y)\\y&={\tfrac {11}{30}}\,(Z+z)\\z&={\tfrac {13}{42}}\,(W+w)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c70b4223542cbecf962cf00f64f4d6d62ac146d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.559ex; margin-bottom: -0.279ex; width:17.901ex; height:26.843ex;" alt="{\displaystyle {\begin{aligned}W&=\;{\tfrac {5}{6}}X+Z\\X&={\tfrac {9}{20}}Y+Z\\Y&={\tfrac {13}{42}}W+Z\\w&={\tfrac {7}{12}}\,(X+x)\\x&={\tfrac {9}{20}}\,(Y+y)\\y&={\tfrac {11}{30}}\,(Z+z)\\z&={\tfrac {13}{42}}\,(W+w)\end{aligned}}}" loading="lazy"></dd></dl>

Die Gesamtzahl der Rinder ist dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W+X+Y+Z+w+x+y+z}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W+X+Y+Z+w+x+y+z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fdd82f39130012baaf93b00e5ea7336223ff37c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.989ex; height:2.509ex;" alt="{\displaystyle W+X+Y+Z+w+x+y+z}" loading="lazy">.

In der schwierigeren Form werden zusätzlich die Nebenbedingungen:

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<annotation encoding="application/x-tex">{\displaystyle W+X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44b4a6cd4a9d6ad36229b10a055e474009a53e93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.256ex; height:2.343ex;" alt="{\displaystyle W+X}" loading="lazy"></td>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"> Quadratzahl</td>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle =m^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9e1984ee7f04da12de4d06e668d3c9144268644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.548ex; height:2.676ex;" alt="{\displaystyle =m^{2}}" loading="lazy"> und

</td></tr>

<tr>

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<annotation encoding="application/x-tex">{\displaystyle Y+Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e274fa1ffcb58e0b7d5a15a3f70a8c7f4ccfd974.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.294ex; height:2.343ex;" alt="{\displaystyle Y+Z}" loading="lazy"></td>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"> Dreieckszahl</td>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle ={\tfrac {1}{2}}(n+1)n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7523ac57a10c09fcd2be9c37abeefaadc79823f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.713ex; height:3.509ex;" alt="{\displaystyle ={\tfrac {1}{2}}(n+1)n}" loading="lazy">.

</td></tr></tbody></table></dd></dl>

verlangt (für ganze Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle m,n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6568e95b6bf8f39b7fd2c9b52b7b00ee124c6250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.469ex; height:2.009ex;" alt="{\displaystyle m,n}" loading="lazy">). Zur Lösungsmethode siehe auch den Artikel über die <a href="Pellsche_Gleichung" title="Pellsche Gleichung">Pellsche Gleichung</a>.

<div class="mw-heading mw-heading2"><h2 id="Detaillierte_Lösung_des_ersten_leichteren_Teils_der_Aufgabe">Detaillierte Lösung des ersten leichteren Teils der Aufgabe</h2></div>

Wenn man die ersten drei Gleichungen geeignet umformt und die Brüche wegbringt, erhält man das folgende Gleichungssystem:

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<annotation encoding="application/x-tex">{\displaystyle 6W}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/933de965991202e492d692e1cdb61a40b0063654.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.598ex; height:2.176ex;" alt="{\displaystyle 6W}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 5X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2273f74d8f461ceabdd0a17672f6cf42d3e7bd65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.142ex; height:2.176ex;" alt="{\displaystyle 5X}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 6Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1bd437350794157af48fbc4962219fd22a10ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.843ex; height:2.176ex;" alt="{\displaystyle 6Z}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 20X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1913d3f202fa52cf2dd40c011cae64ba7252205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.305ex; height:2.176ex;" alt="{\displaystyle 20X}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ae378b108decca9ac4461768eb08507ef7c3983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.936ex; height:2.176ex;" alt="{\displaystyle 9Y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 20Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f25308d5c4d3a978ebe04e065aa32372ff7932.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.005ex; height:2.176ex;" alt="{\displaystyle 20Z}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 13W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eacf06633e38c1dd7ed04cbdbb6a1ec333b8e56e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.76ex; height:2.176ex;" alt="{\displaystyle 13W}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 42Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f78a28990ee97af0f386bb0d98ba140d5226cbba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.098ex; height:2.176ex;" alt="{\displaystyle 42Y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 42Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/714dffca53d5182f6ecd23a69e0db2c7f7d1d708.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.005ex; height:2.176ex;" alt="{\displaystyle 42Z}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Drei Gleichungen mit vier Unbekannten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W,X,Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W,X,Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6527a29eb0a5ea92d864db91ac5dea62d578dc25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.256ex; height:2.509ex;" alt="{\displaystyle W,X,Y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> haben im Normalfall unendlich viele Lösungen, das Gleichungssystem ist also unterbestimmt. Eine Unbekannte ist frei wählbar. Setzt man in diesem Fall die Unbekannte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> als bekannt voraus, so erhält man folgende Lösungen:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {742}{99\cdot 3}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc73a41ac16f43839f3c52b08f1048fc9205781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.44ex; height:3.843ex;" alt="{\displaystyle {\tfrac {742}{99\cdot 3}}Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2226}{891}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1022a35686cecec88e8d25b1ce71a21a2a1d9dd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.804ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2226}{891}}Z}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {178}{99}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ae927348e8249d3211a7aade7aaac0953c876e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.982ex; height:3.843ex;" alt="{\displaystyle {\tfrac {178}{99}}Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1602}{891}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdad7a509a96e3e30856c442c394adc3ca39e827.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.804ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1602}{891}}Z}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1580}{99\cdot 9}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13714c9a06eef6325cabcbb423579ae0014fecf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.804ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1580}{99\cdot 9}}Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1580}{891}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd7b7c41a53df5eb3d1b7303019dde1c71f20c4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.804ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1580}{891}}Z}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Weil aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W,X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W,X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9bb19399397f4cc8e6723248cfa3669d5968aaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.449ex; height:2.509ex;" alt="{\displaystyle W,X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> ganzzahlig sein müssen (es handelt sich immerhin um eine gewisse Anzahl von Rindern), muss auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{891}}Z}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{891}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/793b01c5d03bfc1b6c6a8e68dbac746851df32a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.982ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{891}}Z}" loading="lazy"> ganzzahlig sein. Somit muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> ein Vielfaches von 891 sein. Sei also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z:=891\cdot s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z:=891\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20de23f33c78f3e4ac20b6f0cb890f26474f6028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.683ex; height:2.176ex;" alt="{\displaystyle Z:=891\cdot s}" loading="lazy"> mit ganzzahligem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fde27eac1945df21b2ef2f5a15859e4e9c288698.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {N} }" loading="lazy">. Dann erhält man folgende Lösungen:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2226}{891}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1022a35686cecec88e8d25b1ce71a21a2a1d9dd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.804ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2226}{891}}Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2226\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db7d9b0f9f0336674a33eb2b286f5ec60d747486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.419ex; height:2.176ex;" alt="{\displaystyle 2226\cdot s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1602}{891}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdad7a509a96e3e30856c442c394adc3ca39e827.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.804ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1602}{891}}Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1602\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b6d85556d3e4b110fcf6d2d6c0ae02b63178681.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.419ex; height:2.176ex;" alt="{\displaystyle 1602\cdot s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1580}{891}}Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd7b7c41a53df5eb3d1b7303019dde1c71f20c4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.804ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1580}{891}}Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1580\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3705c83a28df58de2b73457e8460a8e80234177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.419ex; height:2.176ex;" alt="{\displaystyle 1580\cdot s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 891\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f32f281ee1688f877a63999aa06743966d13a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.257ex; height:2.176ex;" alt="{\displaystyle 891\cdot s}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Betrachtet man die anderen vier Gleichungen, bringt die Brüche weg und formt sie geeignet um, erhält man folgendes Gleichungssystem:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 12w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b04f520a01c7bbdc9e86b72255252716f1d1e14a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.989ex; height:2.176ex;" alt="{\displaystyle 12w}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 7x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ca0b9494d351fbeeeea05ad8190031239f75c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 7x}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 7X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/878490e6890564c117421432a3b45f50306c16a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.142ex; height:2.176ex;" alt="{\displaystyle 7X}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 20x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8480758089e1fb00d14b28afdb303a6d26e9d14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.176ex;" alt="{\displaystyle 20x}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3b698b511361cb92ff2275690357d716bd36e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.318ex; height:2.509ex;" alt="{\displaystyle 9y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ae378b108decca9ac4461768eb08507ef7c3983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.936ex; height:2.176ex;" alt="{\displaystyle 9Y}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 30y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99d023980e516649f0905a7512cf2cd0e2f22d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.48ex; height:2.509ex;" alt="{\displaystyle 30y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 11z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6757d4cbe5c8ce363ef076f5d9d8c41a80e9aad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.413ex; height:2.176ex;" alt="{\displaystyle 11z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 11Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55921f5ace359158d048bd95943f9c60c3c89948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.005ex; height:2.176ex;" alt="{\displaystyle 11Z}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 13w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d9bd86ad4ae6b2e6e9bbe87e1ea9a5a50b8f6ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.989ex; height:2.176ex;" alt="{\displaystyle 13w}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 42z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6919a8d98505254c631fd9d7843e007adaf4186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.413ex; height:2.176ex;" alt="{\displaystyle 42z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 13W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eacf06633e38c1dd7ed04cbdbb6a1ec333b8e56e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.76ex; height:2.176ex;" alt="{\displaystyle 13W}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Setzt man nun die Ergebnisse des ersten Gleichungssystems ein, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=2226\cdot s,\ X=1602\cdot s,\ Y=1580\cdot s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=2226\cdot s,\ X=1602\cdot s,\ Y=1580\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1335afc22f4593c652671026b27655b43429ff08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.971ex; height:2.509ex;" alt="{\displaystyle W=2226\cdot s,\ X=1602\cdot s,\ Y=1580\cdot s}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=891\cdot s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z=891\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/328a9772f97386cf09520e3adea58b7cc882d2b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.036ex; height:2.176ex;" alt="{\displaystyle Z=891\cdot s}" loading="lazy">, so erhält man folgendes Gleichungssystem:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 12w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b04f520a01c7bbdc9e86b72255252716f1d1e14a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.989ex; height:2.176ex;" alt="{\displaystyle 12w}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 7x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ca0b9494d351fbeeeea05ad8190031239f75c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 7x}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 11214\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d34052dacec23cd49641045cef1422d573228544.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.582ex; height:2.176ex;" alt="{\displaystyle 11214\cdot s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 20x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8480758089e1fb00d14b28afdb303a6d26e9d14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.176ex;" alt="{\displaystyle 20x}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3b698b511361cb92ff2275690357d716bd36e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.318ex; height:2.509ex;" alt="{\displaystyle 9y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 14220\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72d40aea1b7d13273cca49ae339848a04e393ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.582ex; height:2.176ex;" alt="{\displaystyle 14220\cdot s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 30y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99d023980e516649f0905a7512cf2cd0e2f22d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.48ex; height:2.509ex;" alt="{\displaystyle 30y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 11z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6757d4cbe5c8ce363ef076f5d9d8c41a80e9aad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.413ex; height:2.176ex;" alt="{\displaystyle 11z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9801\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b758c5516d32bbe9a0a9f188c8c5dfc71614959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.419ex; height:2.176ex;" alt="{\displaystyle 9801\cdot s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 13w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d9bd86ad4ae6b2e6e9bbe87e1ea9a5a50b8f6ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.989ex; height:2.176ex;" alt="{\displaystyle 13w}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 42z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6919a8d98505254c631fd9d7843e007adaf4186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.413ex; height:2.176ex;" alt="{\displaystyle 42z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 28938\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d8013223681697a94eb505558ed7ca3a41911a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.582ex; height:2.176ex;" alt="{\displaystyle 28938\cdot s}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Da ja das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> frei wählbar ist, handelt es sich somit um ein eindeutig lösbares Gleichungssystem mit 4 Gleichungen und 4 Unbekannten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w,x,y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w,x,y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ecaddffff3f1b458e160f8f152381aadc89e948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.217ex; height:2.009ex;" alt="{\displaystyle w,x,y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">. Man erhält folgende Lösungen:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {93682680}{4657\cdot 13}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dd04c0078e6687c30a27068c2cab1b1e3f64cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.502ex; height:3.676ex;" alt="{\displaystyle {\tfrac {93682680}{4657\cdot 13}}s}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {7206360}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c648444b86cfee9360200a6a511abc40f093592a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.843ex;" alt="{\displaystyle {\tfrac {7206360}{4657}}s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {97864920}{4657\cdot 20}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38808feaa93eaa7c132767ec9c3ba3247f565351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.502ex; height:3.843ex;" alt="{\displaystyle {\tfrac {97864920}{4657\cdot 20}}s}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {4893246}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/578c095f446cc74092480a986c3d43028763174e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.843ex;" alt="{\displaystyle {\tfrac {4893246}{4657}}s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {105474600}{4657\cdot 30}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4810577a8937f60083948a9868c02a662259f539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.324ex; height:3.843ex;" alt="{\displaystyle {\tfrac {105474600}{4657\cdot 30}}s}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3515820}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/179f4fe76ca087d7f6ce2e4734d5ef4506d7dfe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3515820}{4657}}s}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {5439213}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ce9d4ecd68e9e825ab281a9b380a85f560ba329.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.843ex;" alt="{\displaystyle {\tfrac {5439213}{4657}}s}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Wieder müssen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w,x,y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w,x,y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ecaddffff3f1b458e160f8f152381aadc89e948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.217ex; height:2.009ex;" alt="{\displaystyle w,x,y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> ganzzahlig sein, weil es sich ja um die Anzahlen von Rindern handelt. Somit muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4657}}s}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ba537405cefb69df404ffb7e22c60651a3d97d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.215ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{4657}}s}" loading="lazy"> ganzzahlig sein. Also muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> ein Vielfaches von 4657 sein. Sei also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s:=4657\cdot k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s:=4657\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ef92fca1cb49e463d686a3cd60490615fcbfcca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.376ex; height:2.176ex;" alt="{\displaystyle s:=4657\cdot k}" loading="lazy"> mit ganzzahligem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy">. Dann erhält man die folgenden Lösungen:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {7206360}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c648444b86cfee9360200a6a511abc40f093592a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.843ex;" alt="{\displaystyle {\tfrac {7206360}{4657}}s}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 7206360\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f52867a50676f01683e408772426ba55a8617e17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.028ex; height:2.176ex;" alt="{\displaystyle 7206360\cdot k}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {4893246}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/578c095f446cc74092480a986c3d43028763174e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.843ex;" alt="{\displaystyle {\tfrac {4893246}{4657}}s}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4893246\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97dd989715875ed4e5355c3ef09b067d291530e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.028ex; height:2.176ex;" alt="{\displaystyle 4893246\cdot k}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3515820}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/179f4fe76ca087d7f6ce2e4734d5ef4506d7dfe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3515820}{4657}}s}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3515820\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/078e0d96322492a4771da92587f205bed6fb4ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.028ex; height:2.176ex;" alt="{\displaystyle 3515820\cdot k}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {5439213}{4657}}s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ce9d4ecd68e9e825ab281a9b380a85f560ba329.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.68ex; height:3.843ex;" alt="{\displaystyle {\tfrac {5439213}{4657}}s}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5439213\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/417c97e4b5d41f895dd0a104f28fdd6e1826daef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.028ex; height:2.176ex;" alt="{\displaystyle 5439213\cdot k}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Damit hat der erste und leichtere Teil des archimedischen Rinderproblems (ohne die beiden zusätzlichen Bedingungen) die folgende Lösung (es kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy"> beliebig ganzzahlig gewählt werden):

<td>Die Anzahl der weißen Bullen ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=2226\cdot s=10366482\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/227946b5d0e23314b0cd263bb2925e232e9ac695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:28.242ex; height:2.176ex;" alt="{\displaystyle W=2226\cdot s=10366482\cdot k}" loading="lazy">.

</td></tr>

<tr>

<td>Die Anzahl der schwarzen Bullen ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=1602\cdot s=\;\;7460514\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ecc061654052ed1af17fd5f542c365e71e84aaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.914ex; height:2.176ex;" alt="{\displaystyle X=1602\cdot s=\;\;7460514\cdot k}" loading="lazy">.

</td></tr>

<tr>

<td>Die Anzahl der gefleckten Bullen ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=1580\cdot s=\;\;7358060\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85137e5b5c37fa1441151a34671125faee4ef5ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.707ex; height:2.176ex;" alt="{\displaystyle Y=1580\cdot s=\;\;7358060\cdot k}" loading="lazy">.

</td></tr>

<tr>

<td>Die Anzahl der gelben Bullen ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z=\;\;891\cdot s=\;\;4149387\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79b96c9c07db414fe18dc7c76111df2d3d5b02a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.742ex; height:2.176ex;" alt="{\displaystyle Z=\;\;891\cdot s=\;\;4149387\cdot k}" loading="lazy">.

</td></tr>

<tr>

<td>Die Anzahl der weißen Kühe ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=7206360\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89027b46d74edeb4483dca2a4c11d2f88fe9b0de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.79ex; height:2.176ex;" alt="{\displaystyle w=7206360\cdot k}" loading="lazy">.

</td></tr>

<tr>

<td>Die Anzahl der schwarzen Kühe ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=4893246\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c31a436d5be7c62ac46ee5cb15e21ce0ddf77a96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.456ex; height:2.176ex;" alt="{\displaystyle x=4893246\cdot k}" loading="lazy">.

</td></tr>

<tr>

<td>Die Anzahl der gefleckten Kühe ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=3515820\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54523c5f84f42e8a7defe001ae9577648a73c70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.281ex; height:2.509ex;" alt="{\displaystyle y=3515820\cdot k}" loading="lazy">.

</td></tr>

<tr>

<td>Die Anzahl der gelben Kühe ist</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=5439213\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05713f0101f5cf62fb46e5e2502b89d896b32912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.214ex; height:2.176ex;" alt="{\displaystyle z=5439213\cdot k}" loading="lazy">.

</td></tr></tbody></table></dd></dl>

Die Gesamtzahl der Rinder beträgt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ce34fe991668e617e01fe82e35a39fd98468f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:48.278ex; height:2.509ex;" alt="{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Detaillierte_kleinste_Lösung_des_zweiten_schwierigeren_Teils_der_Aufgabe">Detaillierte kleinste Lösung des zweiten schwierigeren Teils der Aufgabe</h2></div>

Nun zur Lösung des schwierigeren zweiten Teils des Problems:

Setzt man in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W+X=m^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W+X=m^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b55298a380ab222673b8479a8b461e1bcfd11a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.449ex; height:2.843ex;" alt="{\displaystyle W+X=m^{2}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=10366482\cdot k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=10366482\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25379659d0c046fe39321d4ce434d92cec08bc3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.724ex; height:2.176ex;" alt="{\displaystyle W=10366482\cdot k}" loading="lazy"> und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=7460514\cdot k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=7460514\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f63961ef1fb0cf27ca365ea93cbf42b7cbeb0a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.106ex; height:2.176ex;" alt="{\displaystyle X=7460514\cdot k}" loading="lazy"> ein, so erhält man

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 10366482\cdot k+7460514\cdot k=17826996\cdot k=m^{2}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfdd30ec4240ff5630d4804e30cea6c3405699ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:48.186ex; height:2.843ex;" alt="{\displaystyle 10366482\cdot k+7460514\cdot k=17826996\cdot k=m^{2}.}" loading="lazy"></dd></dl>

Mit der <a href="Primfaktorzerlegung" title="Primfaktorzerlegung">Primfaktorzerlegung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17826996=2^{2}\cdot 3\cdot 11\cdot 29\cdot 4657}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 17826996=2^{2}\cdot 3\cdot 11\cdot 29\cdot 4657}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d17d203d5b1f59c792a25f42a9014461c1d971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:31.793ex; height:2.676ex;" alt="{\displaystyle 17826996=2^{2}\cdot 3\cdot 11\cdot 29\cdot 4657}" loading="lazy"> erhält man die Gleichung

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{2}\cdot 3\cdot 11\cdot 29\cdot 4657\cdot k=m^{2}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d005fc9430ba7a36616f8ea5f5b02b6fc77b825f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.125ex; height:2.676ex;" alt="{\displaystyle 2^{2}\cdot 3\cdot 11\cdot 29\cdot 4657\cdot k=m^{2}.}" loading="lazy"></dd></dl>

Damit der linke Teil ein vollständiges Quadrat ist, muss gelten:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=3\cdot 11\cdot 29\cdot 4657\cdot r^{2}=4456749\cdot r^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59edd8e6ab69682ecbdddb67483837eb57a93432.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:38.609ex; height:2.676ex;" alt="{\displaystyle k=3\cdot 11\cdot 29\cdot 4657\cdot r^{2}=4456749\cdot r^{2}}" loading="lazy"></dd></dl>

mit einem ganzzahligen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f7b1b20ccd888e84ebffce5446840cc882b3dc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.567ex; height:2.176ex;" alt="{\displaystyle r\in \mathbb {N} }" loading="lazy">.

Nun betrachtet man die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y+Z={\tfrac {n\cdot (n+1)}{2}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y+Z={\tfrac {n\cdot (n+1)}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed3b7f234ee8645e55b510f21ab95e251405c11f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.038ex; height:4.176ex;" alt="{\displaystyle Y+Z={\tfrac {n\cdot (n+1)}{2}}}" loading="lazy">. Setzt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:=2n+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:=2n+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6eb8bacd45004d86535d5286826261df6fbd484.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.644ex; height:2.343ex;" alt="{\displaystyle h:=2n+1}" loading="lazy">, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\tfrac {h-1}{2}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n={\tfrac {h-1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31d9aaf0cabe92bee1b551d039d1ca391eb49577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.377ex; height:3.676ex;" alt="{\displaystyle n={\tfrac {h-1}{2}}}" loading="lazy">, dann kann man das auch schreiben als

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y+Z={\tfrac {1}{2}}\cdot {\tfrac {h-1}{2}}\cdot {\tfrac {h+1}{2}}={\tfrac {1}{8}}(h^{2}-1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41858874bfe8ef0e2169a4bd4ccefe99fca5e9f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.138ex; height:3.843ex;" alt="{\displaystyle Y+Z={\tfrac {1}{2}}\cdot {\tfrac {h-1}{2}}\cdot {\tfrac {h+1}{2}}={\tfrac {1}{8}}(h^{2}-1)}" loading="lazy"></dd></dl>

bzw. als

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+8(Y+Z)=h^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97bcdb66a18e259a4fb134f337bbadc4e4906239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.76ex; height:3.176ex;" alt="{\displaystyle 1+8(Y+Z)=h^{2}}" loading="lazy">.</dd></dl>

Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> ganzzahlig ist, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> offensichtlich ungerade.

Setzt man die Zwischenergebnisse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=1580\cdot 4657\cdot k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=1580\cdot 4657\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/524d4470149755b6bf40f83c0a7b4abb6e431022.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.741ex; height:2.176ex;" alt="{\displaystyle Y=1580\cdot 4657\cdot k}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=891\cdot 4657\cdot k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z=891\cdot 4657\cdot k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84db066e2f66dcc6cc538be82e10048d9a851a96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.485ex; height:2.176ex;" alt="{\displaystyle Z=891\cdot 4657\cdot k}" loading="lazy"> ein, so erhält man wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1580+891=2471=7\cdot 353}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1580+891=2471=7\cdot 353}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09ea916de7ac68a40220903f01f17d123b4c991e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.153ex; height:2.343ex;" alt="{\displaystyle 1580+891=2471=7\cdot 353}" loading="lazy"> für die Summe

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<mtd>

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<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Y+Z&=7\cdot 353\cdot 4657\cdot k\\&=3\cdot 7\cdot 11\cdot 29\cdot 353\cdot 4657^{2}\cdot r^{2}\\&=2364747\cdot 4657^{2}\cdot r^{2}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ab2bbe0be8c009455cc9dd4c6aa195c2956b4a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:38.488ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}Y+Z&=7\cdot 353\cdot 4657\cdot k\\&=3\cdot 7\cdot 11\cdot 29\cdot 353\cdot 4657^{2}\cdot r^{2}\\&=2364747\cdot 4657^{2}\cdot r^{2}\end{aligned}}}" loading="lazy"></dd></dl>

und daraus als Beziehung zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> die Gleichung

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<msup>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+8\cdot 2364747\cdot 4657^{2}\cdot r^{2}=h^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daa52678cfa803d255431ad5263a8380e84fa38f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:31.638ex; height:2.843ex;" alt="{\displaystyle 1+8\cdot 2364747\cdot 4657^{2}\cdot r^{2}=h^{2}}" loading="lazy">.</dd></dl>

Weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\cdot 2364747\cdot 4657^{2}=410286423278424}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 8\cdot 2364747\cdot 4657^{2}=410286423278424}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe0a07028fe4188d90541d6009e1d5ccfef18d4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:38.897ex; height:2.676ex;" alt="{\displaystyle 8\cdot 2364747\cdot 4657^{2}=410286423278424}" loading="lazy"> ist, ergibt sich die folgende <a href="Pellsche_Gleichung" title="Pellsche Gleichung">Pellsche Gleichung</a>:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb933051a807498abc47674630b639c66447e0ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.713ex; height:2.843ex;" alt="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}" loading="lazy"></dd></dl>

Berücksichtigt man die Faktorzerlegung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 410286423278424=4729494\cdot (2\cdot 4657)^{2},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 410286423278424=4729494\cdot (2\cdot 4657)^{2},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6f07a36f5c2d21ed8a4a0938513190ec288cff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.353ex; height:3.176ex;" alt="{\displaystyle 410286423278424=4729494\cdot (2\cdot 4657)^{2},}" loading="lazy"> so erhält man die etwas einfacher zu lösende Pellsche Gleichung:

<msup>

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</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05400d8ff2ad30449aae679bc4e767bcb0e0a69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.714ex; height:3.176ex;" alt="{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}" loading="lazy"></dd></dl>

Diese Pellsche Gleichung hat, weil 4729494 keine <a href="Quadratzahl" title="Quadratzahl">Quadratzahl</a> ist, unendlich viele Lösungen.

Man löst sie mit der <a href="Kettenbruchentwicklung" class="mw-redirect" title="Kettenbruchentwicklung">Kettenbruchentwicklung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {4729494}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {4729494}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/824e274f1ff9d1b9793d1785809b01a3057f828f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.073ex; height:3.009ex;" alt="{\displaystyle {\sqrt {4729494}}}" loading="lazy">, die mit einer Periodenlänge von 92 und somit mit einer Gesamtlänge von 93 wie folgt lautet:

<mtr>

<mtd>

<msqrt>

</msqrt>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sqrt {4729494}}=[2174;{\overline {1,2,1,5,2,25,3,1,1,1,1,1,1,15,1,2,16,1,2,1,1,8,6,1,21,1,1,3,1,1,1,2,2,6,1,1,5,1,17,1,1,47,3,}}\\{\overline {1,1,6,1,1,3,47,1,1,17,1,5,1,1,6,2,2,1,1,1,3,1,1,21,1,6,8,1,1,2,1,16,2,1,15,1,1,1,1,1,1,3,25,2,5,1,2,1,4348}}]\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40303219d90e54740743985326aaf24182dc44d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:121.401ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\sqrt {4729494}}=[2174;{\overline {1,2,1,5,2,25,3,1,1,1,1,1,1,15,1,2,16,1,2,1,1,8,6,1,21,1,1,3,1,1,1,2,2,6,1,1,5,1,17,1,1,47,3,}}\\{\overline {1,1,6,1,1,3,47,1,1,17,1,5,1,1,6,2,2,1,1,1,3,1,1,21,1,6,8,1,1,2,1,16,2,1,15,1,1,1,1,1,1,3,25,2,5,1,2,1,4348}}]\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

Die obige Pellsche Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05400d8ff2ad30449aae679bc4e767bcb0e0a69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.714ex; height:3.176ex;" alt="{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}" loading="lazy"> hat die folgende kleinste (Minimal-)Lösung:

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<mtd>

<msub>

</mrow>

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<mtd>

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</mrow>

</msup>

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</mtr>

<mtr>

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<msub>

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</msub>

<msub>

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</msub>

</mtd>

<mtd>

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<mtd>

<msup>

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</mtr>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{0}=&109931986732829734979866232821433543901088049&\approx 1{,}099\cdot 10^{44}\\g_{0}:=2\cdot 4657r_{0}=&50549485234315033074477819735540408986340&\approx 5{,}055\cdot 10^{40}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b4bcfa80dbb844884ad7a09108a74f39bed3434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:89.294ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}h_{0}=&109931986732829734979866232821433543901088049&\approx 1{,}099\cdot 10^{44}\\g_{0}:=2\cdot 4657r_{0}=&50549485234315033074477819735540408986340&\approx 5{,}055\cdot 10^{40}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

Somit ist aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}={\tfrac {g_{0}}{2\cdot 4657}}={\tfrac {25274742617157516537238909867770204493170}{4657}}\notin \mathbb {N} }">

<msub>

</mrow>

</msub>

<mfrac>

<msub>

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</msub>

<mrow>

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</mfrac>

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</mrow>

<mfrac>

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</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{0}={\tfrac {g_{0}}{2\cdot 4657}}={\tfrac {25274742617157516537238909867770204493170}{4657}}\notin \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43fcd8c86e6dddabbc49304561fa8d790258f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:52.759ex; height:3.843ex;" alt="{\displaystyle r_{0}={\tfrac {g_{0}}{2\cdot 4657}}={\tfrac {25274742617157516537238909867770204493170}{4657}}\notin \mathbb {N} }" loading="lazy"> nicht ganzzahlig und deswegen ist die obige Minimallösung der Pellschen Gleichung noch immer nicht die gesuchte Lösung des Rinderproblems.

Es muss das gesuchte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> ein Teiler von einem gewissen (kleinstmöglichen) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> sein, das die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot 4657}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\cdot 4657}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22e725e4d80a3d29b414f7c9016bf0b148dea92e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.491ex; height:2.176ex;" alt="{\displaystyle 2\cdot 4657}" loading="lazy"> als Teiler hat.

Wenn man die Minimallösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h_{0},g_{0})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (h_{0},g_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00fce1b8a915ebf090109b4742439982b56cb11b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.4ex; height:2.843ex;" alt="{\displaystyle (h_{0},g_{0})}" loading="lazy"> der obigen Pellschen Gleichung hat, muss man nur noch die folgenden beiden Iterationsformeln anwenden, mit denen man alle weiteren (unendlich vielen) Lösungen der Pellschen Gleichung erhält (siehe <a href="Pellsche_Gleichung#Generieren_weiterer_Lösungen" title="Pellsche Gleichung">Pellsche Gleichung#Generieren weiterer Lösungen</a>):

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

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<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{i+1}&=&h_{0}\cdot h_{i}+4729494\cdot g_{0}\cdot g_{i}\\g_{i+1}&=&g_{0}\cdot h_{i}+h_{0}\cdot g_{i}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79d228c43999ee958b932860ba74c81249073083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.111ex; margin-bottom: -0.227ex; width:36.708ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}h_{i+1}&=&h_{0}\cdot h_{i}+4729494\cdot g_{0}\cdot g_{i}\\g_{i+1}&=&g_{0}\cdot h_{i}+h_{0}\cdot g_{i}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

mit obigem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08c1c908b03c3f63383c7199465c7fd0b105030f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32d13273b9af4564fa2c421c96d039c414db8628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{0}}" loading="lazy">.

Der erste Iterationsschritt ist

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{1}&=&h_{0}\cdot h_{0}+4729494\cdot g_{0}\cdot g_{0}\\g_{1}&=&g_{0}\cdot h_{0}+h_{0}\cdot g_{0}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39a080573e43b8f9df8ddaaffdc2807ede11a090.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.371ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}h_{1}&=&h_{0}\cdot h_{0}+4729494\cdot g_{0}\cdot g_{0}\\g_{1}&=&g_{0}\cdot h_{0}+h_{0}\cdot g_{0}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

Leider hat auch dieses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}\approx 1{,}1114\cdot 10^{85}}">

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<annotation encoding="application/x-tex">{\displaystyle g_{1}\approx 1{,}1114\cdot 10^{85}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/048e34eb38feea2662c23185dab15b45c4ad358b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.601ex; height:3.009ex;" alt="{\displaystyle g_{1}\approx 1{,}1114\cdot 10^{85}}" loading="lazy"> nicht die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot 4657}">

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<annotation encoding="application/x-tex">{\displaystyle 2\cdot 4657}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22e725e4d80a3d29b414f7c9016bf0b148dea92e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.491ex; height:2.176ex;" alt="{\displaystyle 2\cdot 4657}" loading="lazy"> als Teiler.

Auch mit dem nächsten, dem zweiten Iterationsschritt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{2}&=&h_{0}\cdot h_{1}+4729494\cdot g_{0}\cdot g_{1}\\g_{2}&=&g_{0}\cdot h_{1}+h_{0}\cdot g_{1}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af00e8d833ef6e4e043f07d1d4ef06c6ca1c1feb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.371ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}h_{2}&=&h_{0}\cdot h_{1}+4729494\cdot g_{0}\cdot g_{1}\\g_{2}&=&g_{0}\cdot h_{1}+h_{0}\cdot g_{1}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

erhält man ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}\approx 2{,}44357\cdot 10^{129}}">

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<annotation encoding="application/x-tex">{\displaystyle g_{2}\approx 2{,}44357\cdot 10^{129}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce93646d80109cea09d42ec6a6089c1a72d50014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.586ex; height:3.009ex;" alt="{\displaystyle g_{2}\approx 2{,}44357\cdot 10^{129}}" loading="lazy">, das die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot 4657}">

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<annotation encoding="application/x-tex">{\displaystyle 2\cdot 4657}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22e725e4d80a3d29b414f7c9016bf0b148dea92e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.491ex; height:2.176ex;" alt="{\displaystyle 2\cdot 4657}" loading="lazy"> nicht als Teiler hat.

Erst nach unglaublichen 2329 Iterationsschritten erhält man die folgende gesuchte kleinste Lösung:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h:=h_{2329}&\approx &3{,}765344502347206\cdot 10^{103272}\\g:=g_{2329}&\approx &1{,}731399858951771\cdot 10^{103269}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bbcb179854af8c60cd6ec2303fb2e658c2feefd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.302ex; margin-bottom: -0.203ex; width:46.208ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}h:=h_{2329}&\approx &3{,}765344502347206\cdot 10^{103272}\\g:=g_{2329}&\approx &1{,}731399858951771\cdot 10^{103269}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

Und somit erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\tfrac {g}{2\cdot 4657}}\approx 1{,}858921901386913\cdot 10^{103265}\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle r={\tfrac {g}{2\cdot 4657}}\approx 1{,}858921901386913\cdot 10^{103265}\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b450d23d5d5e363ed9f1b7afc60f7deaabd326e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:45.582ex; height:3.676ex;" alt="{\displaystyle r={\tfrac {g}{2\cdot 4657}}\approx 1{,}858921901386913\cdot 10^{103265}\in \mathbb {N} }" loading="lazy">. Damit kann man auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=4456749\cdot r^{2}\approx 1{,}540070010897761\cdot 10^{206537}}">

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<annotation encoding="application/x-tex">{\displaystyle k=4456749\cdot r^{2}\approx 1{,}540070010897761\cdot 10^{206537}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca849a4e02fd64bdcbaf6c73a860c9836c002d9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.742ex; height:3.009ex;" alt="{\displaystyle k=4456749\cdot r^{2}\approx 1{,}540070010897761\cdot 10^{206537}}" loading="lazy"> errechnen und oben bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W,X,Y,Z,w,x,y}">

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<annotation encoding="application/x-tex">{\displaystyle W,X,Y,Z,w,x,y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1765e5b1e55261527e89f2989c4128d5d463c35f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.222ex; height:2.509ex;" alt="{\displaystyle W,X,Y,Z,w,x,y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> einsetzen.

Die schwierigere Form des archimedischen Rinderproblems (also inklusive der beiden zusätzlichen Bedingungen) hat somit die folgende kleinste Lösung:

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<annotation encoding="application/x-tex">{\displaystyle k\approx 1{,}54007001089776119944598967554\cdot 10^{206537}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2c5f390236db917ebf56690aecec01e28d012ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:48.998ex; height:3.009ex;" alt="{\displaystyle k\approx 1{,}54007001089776119944598967554\cdot 10^{206537}}" loading="lazy"></dd></dl>

<table>

<td>Die Anzahl der weißen Bullen ist</td>

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<annotation encoding="application/x-tex">{\displaystyle W=10366482\cdot k\approx 1{,}5965108046711445314\cdot 10^{206544}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/020d86e47c9111288d7afd63673e43da59305e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.886ex; height:3.009ex;" alt="{\displaystyle W=10366482\cdot k\approx 1{,}5965108046711445314\cdot 10^{206544}}" loading="lazy">.

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<td>Die Anzahl der schwarzen Bullen ist</td>

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<annotation encoding="application/x-tex">{\displaystyle X=\;\;7460514\cdot k\approx 1{,}1489713877282899997\cdot 10^{206544}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d6a3ea1b97984c4e1bbdda6df238a585b2899e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.559ex; height:3.009ex;" alt="{\displaystyle X=\;\;7460514\cdot k\approx 1{,}1489713877282899997\cdot 10^{206544}}" loading="lazy">.

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<td>Die Anzahl der gefleckten Bullen ist</td>

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<annotation encoding="application/x-tex">{\displaystyle Y=\;\;7358060\cdot k\approx 1{,}1331927544386380771\cdot 10^{206544}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d5740ea82c3534e95be4886a2342b07421282c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.352ex; height:3.009ex;" alt="{\displaystyle Y=\;\;7358060\cdot k\approx 1{,}1331927544386380771\cdot 10^{206544}}" loading="lazy">.

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<td>Die Anzahl der gelben Bullen ist</td>

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<annotation encoding="application/x-tex">{\displaystyle Z=\;\;4149387\cdot k\approx 6{,}3903464823090286501\cdot 10^{206543}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9620985b469e782411018a5b5c55277a5114a6b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.259ex; height:3.009ex;" alt="{\displaystyle Z=\;\;4149387\cdot k\approx 6{,}3903464823090286501\cdot 10^{206543}}" loading="lazy">.

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<td>Die Anzahl der weißen Kühe ist</td>

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<annotation encoding="application/x-tex">{\displaystyle w=\;\;7206360\cdot k\approx 1{,}1098298923733190397\cdot 10^{206544}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d4e757dcbfd6c48c66bc01367f80130ffcc8a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.243ex; height:3.009ex;" alt="{\displaystyle w=\;\;7206360\cdot k\approx 1{,}1098298923733190397\cdot 10^{206544}}" loading="lazy">.

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<td>Die Anzahl der schwarzen Kühe ist</td>

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<annotation encoding="application/x-tex">{\displaystyle x=\;\;4893246\cdot k\approx 7{,}5359414205454263981\cdot 10^{206543}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4488350c447bf7cbe7988c36b86fdc619c371529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:52.909ex; height:3.009ex;" alt="{\displaystyle x=\;\;4893246\cdot k\approx 7{,}5359414205454263981\cdot 10^{206543}}" loading="lazy">.

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<td>Die Anzahl der gefleckten Kühe ist</td>

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<annotation encoding="application/x-tex">{\displaystyle y=\;\;3515820\cdot k\approx 5{,}4146089457145667802\cdot 10^{206543}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d708ca882b20feddd09b8fb2d20cb796925402c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:52.734ex; height:3.009ex;" alt="{\displaystyle y=\;\;3515820\cdot k\approx 5{,}4146089457145667802\cdot 10^{206543}}" loading="lazy">.

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<td>Die Anzahl der gelben Kühe ist</td>

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<annotation encoding="application/x-tex">{\displaystyle z=\;\;5439213\cdot k\approx 8{,}3767688241852443869\cdot 10^{206543}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8f31afb70a37568e3b40c04b10046e5bba21c77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:52.667ex; height:3.009ex;" alt="{\displaystyle z=\;\;5439213\cdot k\approx 8{,}3767688241852443869\cdot 10^{206543}}" loading="lazy">.

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Die Gesamtzahl der Rinder beträgt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k\approx 7{,}7602714064868182695\cdot 10^{206544}}">

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<annotation encoding="application/x-tex">{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k\approx 7{,}7602714064868182695\cdot 10^{206544}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7891d05eea5bf41896c6b1981cbb6dd5af763c70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:84.44ex; height:3.009ex;" alt="{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k\approx 7{,}7602714064868182695\cdot 10^{206544}}" loading="lazy">.

Die exakten Ergebnisse für die Anzahl der einzelnen Bullen und Kühe und für die Gesamtzahl der Rinder, also alle 206544 bzw. 206545 Stellen, kann man auf einer Internetseite<a href="#cite_note-10">[10]</a> nachlesen.

Die weißen Bullen, von denen es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> Stück gibt, und die schwarzen Bullen, von denen es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> Stück gibt, kann man quadratisch so anordnen, dass auf jeder Seite des Quadrats <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\sqrt {W+X}}\approx 1{,}656949665016845\cdot 10^{103272}}">

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<annotation encoding="application/x-tex">{\displaystyle m={\sqrt {W+X}}\approx 1{,}656949665016845\cdot 10^{103272}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d266207128fa2b6c87d908a9b19489042bdf9512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.843ex; height:3.176ex;" alt="{\displaystyle m={\sqrt {W+X}}\approx 1{,}656949665016845\cdot 10^{103272}}" loading="lazy"> Bullen stehen.

Die gefleckten Bullen, von denen es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> Stück gibt, und die gelben Bullen, von denen es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> Stück gibt, kann man dreieckig so anordnen, dass auf jeder Seite des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=n_{1}={\tfrac {-1+{\sqrt {1+8(Y+Z)}}}{2}}\approx 1{,}882672251173603\cdot 10^{103272}}">

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<annotation encoding="application/x-tex">{\displaystyle n=n_{1}={\tfrac {-1+{\sqrt {1+8(Y+Z)}}}{2}}\approx 1{,}882672251173603\cdot 10^{103272}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f89dcad538fc5f5229d57577fce5d5cee3154842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:55.334ex; height:4.509ex;" alt="{\displaystyle n=n_{1}={\tfrac {-1+{\sqrt {1+8(Y+Z)}}}{2}}\approx 1{,}882672251173603\cdot 10^{103272}}" loading="lazy"> Bullen stehen.

<div class="mw-heading mw-heading2"><h2 id="Alle_Lösungen_des_zweiten_schwierigeren_Teils_der_Aufgabe">Alle Lösungen des zweiten schwierigeren Teils der Aufgabe</h2></div>

Die weiter oben stehende Pellsche Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb933051a807498abc47674630b639c66447e0ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.713ex; height:2.843ex;" alt="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}" loading="lazy">, die umgeformt wurde zur Pellschen Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05400d8ff2ad30449aae679bc4e767bcb0e0a69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.714ex; height:3.176ex;" alt="{\displaystyle h^{2}-4729494\cdot (2\cdot 4657r)^{2}=1}" loading="lazy">, hat natürlich dieselbe oben schon erhaltene ganzzahlige Minimallösung

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h\approx &\;3{,}765344502347206\cdot 10^{103272}\\r={\frac {g}{2\cdot 4657}}\approx &\;1{,}858921901386913\cdot 10^{103265}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c374ee1dfa1d30fd4b69491b3b02f9795d1c6395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.739ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}h\approx &\;3{,}765344502347206\cdot 10^{103272}\\r={\frac {g}{2\cdot 4657}}\approx &\;1{,}858921901386913\cdot 10^{103265}\end{aligned}}}" loading="lazy">.</dd></dl></dd></dl>

Diese Pellsche Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb933051a807498abc47674630b639c66447e0ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.713ex; height:2.843ex;" alt="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}" loading="lazy"> hat unendlich viele Lösungen. Jede dieser Lösungen ist auch gleichzeitig Lösung des Rinderproblems. Somit hat auch das Rinderproblem unendlich viele Lösungen.

Wenn man die Minimallösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h,r)}">

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<annotation encoding="application/x-tex">{\displaystyle (h,r)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33df6e72d4b5536671e8defe5be12b9aada3844e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.231ex; height:2.843ex;" alt="{\displaystyle (h,r)}" loading="lazy"> der obigen Pellschen Gleichung kennt, muss man wieder die folgenden beiden Iterationsformeln anwenden, mit denen man alle weiteren (unendlich vielen) Lösungen der Pellschen Gleichung erhält:

Der erste Iterationsschritt ist

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{1}=&\;h_{0}\cdot h_{0}+410286423278424\cdot r_{0}\cdot r_{0}\\r_{1}=&\;r_{0}\cdot h_{0}+h_{0}\cdot r_{0}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae5da44bf0eedff8191f6d242e69087125f95cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.55ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}h_{1}=&\;h_{0}\cdot h_{0}+410286423278424\cdot r_{0}\cdot r_{0}\\r_{1}=&\;r_{0}\cdot h_{0}+h_{0}\cdot r_{0}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}:=h}">

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<annotation encoding="application/x-tex">{\displaystyle h_{0}:=h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2a197941ef5f05fdd6d9a5979d92baa1efd3dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.477ex; height:2.509ex;" alt="{\displaystyle h_{0}:=h}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}:=r}">

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<annotation encoding="application/x-tex">{\displaystyle r_{0}:=r}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f644ca46398a4b9fce5ac1169dda45cebb6fc9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.897ex; height:2.009ex;" alt="{\displaystyle r_{0}:=r}" loading="lazy">.

Tatsächlich ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}\approx 2{,}835563844271266\cdot 10^{206545}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{1}\approx 2{,}835563844271266\cdot 10^{206545}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c456c8558ea55673942a9b8d7066b1294b8517a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.906ex; height:3.009ex;" alt="{\displaystyle h_{1}\approx 2{,}835563844271266\cdot 10^{206545}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}\approx 1{,}399896272336006\cdot 10^{206538}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{1}\approx 1{,}399896272336006\cdot 10^{206538}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cad94a489975b8436d5e63ff8a048fecd46a3414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.616ex; height:3.009ex;" alt="{\displaystyle r_{1}\approx 1{,}399896272336006\cdot 10^{206538}}" loading="lazy"> die zweitkleinste Lösung der Pellschen Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb933051a807498abc47674630b639c66447e0ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.713ex; height:2.843ex;" alt="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}" loading="lazy">.

Mit dem nächsten, dem zweiten Iterationsschritt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{2}=&\;h_{0}\cdot h_{1}+410286423278424\cdot r_{0}\cdot r_{1}\\r_{2}=&\;r_{0}\cdot h_{1}+h_{0}\cdot r_{1}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82c1a73129f69f027604b03eca81748012aa92eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.55ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}h_{2}=&\;h_{0}\cdot h_{1}+410286423278424\cdot r_{0}\cdot r_{1}\\r_{2}=&\;r_{0}\cdot h_{1}+h_{0}\cdot r_{1}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

erhält man die drittkleinste Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h_{2},r_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle (h_{2},r_{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04c5b1b0a69e6a14588093485edafc0f62b6842f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.339ex; height:2.843ex;" alt="{\displaystyle (h_{2},r_{2})}" loading="lazy"> der Pellschen Gleichung.

Generell gilt:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{i+1}=&\;h_{0}\cdot h_{i}+410286423278424\cdot r_{0}\cdot r_{i}\\r_{i+1}=&\;r_{0}\cdot h_{i}+h_{0}\cdot r_{i}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b6e1abcacbe50e02f9af72c1056b4b7d794b1b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.111ex; margin-bottom: -0.227ex; width:41.887ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}h_{i+1}=&\;h_{0}\cdot h_{i}+410286423278424\cdot r_{0}\cdot r_{i}\\r_{i+1}=&\;r_{0}\cdot h_{i}+h_{0}\cdot r_{i}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

ergibt alle Lösungspaare <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h_{i+1},r_{i+1})}">

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<annotation encoding="application/x-tex">{\displaystyle (h_{i+1},r_{i+1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90dfdb85aa76ad3313dc28d6f6c399bb26ccc9cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.031ex; height:2.843ex;" alt="{\displaystyle (h_{i+1},r_{i+1})}" loading="lazy"> der Pellschen Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb933051a807498abc47674630b639c66447e0ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.713ex; height:2.843ex;" alt="{\displaystyle h^{2}-410286423278424\cdot r^{2}=1}" loading="lazy">.

Interessant sind aber vor allem die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}" loading="lazy">. Damit kann man nämlich wieder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}=4456749\cdot r_{i}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{i}=4456749\cdot r_{i}^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdb9f75da24e41bbfb3d24b54b17b658a2be5a58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.029ex; height:3.176ex;" alt="{\displaystyle k_{i}=4456749\cdot r_{i}^{2}}" loading="lazy"> errechnen und oben bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W,X,Y,Z,w,x,y}">

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<annotation encoding="application/x-tex">{\displaystyle W,X,Y,Z,w,x,y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1765e5b1e55261527e89f2989c4128d5d463c35f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.222ex; height:2.509ex;" alt="{\displaystyle W,X,Y,Z,w,x,y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> einsetzen. So erhält man alle weiteren Lösungen des Rinderproblems.

<div class="mw-heading mw-heading2"><h2 id="Alle_Lösungen_des_zweiten_schwierigeren_Teils_der_Aufgabe_–_Alternative_Formeln">Alle Lösungen des zweiten schwierigeren Teils der Aufgabe – Alternative Formeln</h2></div>

Wenn man alle unendlich vielen Lösungen dieses Rinderproblems wissen will, so bieten sich auch die folgenden Formeln an, die der Mathematiker H. W. Lenstra Jr. auf<a href="#cite_note-11">[11]</a> veröffentlicht hat.

Sei

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle c:=300426607914281713365\cdot {\sqrt {609}}+84129507677858393258\cdot {\sqrt {7766}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63b08e680a69105a37e97368cd2107a19f977dca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:70.62ex; height:3.009ex;" alt="{\displaystyle c:=300426607914281713365\cdot {\sqrt {609}}+84129507677858393258\cdot {\sqrt {7766}}}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle k_{j}:={\frac {(c^{4658\cdot j}-c^{-4658\cdot j})^{2}}{368238304}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca762ebb67750e3bd32f1e47a8571f588ae70596.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.009ex; height:6.009ex;" alt="{\displaystyle k_{j}:={\frac {(c^{4658\cdot j}-c^{-4658\cdot j})^{2}}{368238304}}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,2,3,\dotsc }">

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<annotation encoding="application/x-tex">{\displaystyle j=1,2,3,\dotsc }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bfa6276fe5656cd0967e8f2043407165579060f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:13.396ex; height:2.509ex;" alt="{\displaystyle j=1,2,3,\dotsc }" loading="lazy">

Die kleinste Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/376315fd4983f01dada5ec2f7bebc48455b14a66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}" loading="lazy"> ist die obige errechnete kleinste Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> des Rinderproblems.

Die schwierige Form des archimedischen Rinderproblems (inklusive der beiden zusätzlichen Bedingungen) hat die folgende <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">-kleinste Lösung (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b54b0f8d5a5eb196ce103460f69c9eb3ec9393fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.504ex; height:2.509ex;" alt="{\displaystyle j\in \mathbb {N} }" loading="lazy"> kann beliebig positiv ganzzahlig gewählt werden):

Die Anzahl der weißen Bullen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=10366482\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=10366482\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6662edd1d6c33a93ccf191241bafecae9e2c05d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.633ex; height:2.843ex;" alt="{\displaystyle W=10366482\cdot k_{j}}" loading="lazy">.

Die Anzahl der schwarzen Bullen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=7460514\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=7460514\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fddd7cc5a49a1baff3426be14247deaa8946878f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.016ex; height:2.843ex;" alt="{\displaystyle X=7460514\cdot k_{j}}" loading="lazy">.

Die Anzahl der gefleckten Bullen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=7358060\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=7358060\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cba1944fd1f621c0204219ba74d858efe12a3883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.809ex; height:2.843ex;" alt="{\displaystyle Y=7358060\cdot k_{j}}" loading="lazy">.

Die Anzahl der gelben Bullen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=4149387\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z=4149387\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b4c79260b5972593cd31a06f006d7aa1d5994c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.716ex; height:2.843ex;" alt="{\displaystyle Z=4149387\cdot k_{j}}" loading="lazy">.

Die Anzahl der weißen Kühe ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=7206360\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=7206360\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f0db35100778f96f056935e8126ed2679cbab9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.7ex; height:2.843ex;" alt="{\displaystyle w=7206360\cdot k_{j}}" loading="lazy">.

Die Anzahl der schwarzen Kühe ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=4893246\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=4893246\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e476a12e0f4dd472320b43075aa2969fa0e4374c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.365ex; height:2.843ex;" alt="{\displaystyle x=4893246\cdot k_{j}}" loading="lazy">.

Die Anzahl der gefleckten Kühe ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=3515820\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=3515820\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2352335fe2e3829c65e7e93a256f406875e45891.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.191ex; height:2.843ex;" alt="{\displaystyle y=3515820\cdot k_{j}}" loading="lazy">.

Die Anzahl der gelben Kühe ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=5439213\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=5439213\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d33ecef08357e0b1619d24250d4397ef3fe69b37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.124ex; height:2.843ex;" alt="{\displaystyle z=5439213\cdot k_{j}}" loading="lazy">.

Die Gesamtzahl der Rinder beträgt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ef98b581ed3cd5948b63c400cbca580e37b7cd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.187ex; height:2.843ex;" alt="{\displaystyle W+X+Y+Z+w+x+y+z=50389082\cdot k_{j}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1993_Schreiber-1">↑ <a href="#cite_ref-1993_Schreiber_1-0">a</a> <a href="#cite_ref-1993_Schreiber_1-1">b</a> Peter Schreiber: <style data-mw-deduplicate="TemplateStyles:r261891140">

/* start https://de.wikipedia.org/ */

.mw-parser-output .webarchiv-memento a{color:inherit}

/* end https://de.wikipedia.org/ */

</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131203214323/http://poncelet.math.nthu.edu.tw/disk5/js/history/cows.pdf">A Note on the Cattle Problem of Archimedes.</a> (<a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a> vom 3. Dezember 2013 im <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>). (PDF; 131 kB). In: Historia Mathematica. 20, 1993, S. 304–306.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Gotthold Ephraim Lessing: Zur Geschichte und Litteratur. Aus den Schätzen der Herzoglichen Bibliothek zu Wolfenbüttel. Zweiter Beitrag. Braunschweig 1773, S. <a rel="nofollow" class="external text" href="http://diglib.hab.de/wdb.php?dir=drucke/lo-4637-1b-2s&pointer=166">421–446</a>.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a rel="nofollow" class="external free" href="http://users.ntua.gr/dimour/greats/Archimedes/index.html">http://users.ntua.gr/dimour/greats/Archimedes/index.html</a>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Vgl. u. a. Jacob Struve, Karl Ludwig Struve: Altes Griechisches Epigramm mathematischen Inhalts, mathematisch und kritisch behandelt. Altona 1821.

</li>

<li id="cite_note-:0-5">↑ <a href="#cite_ref-:0_5-0">a</a> <a href="#cite_ref-:0_5-1">b</a> Bernhard Krumbiegel, August Amthor: <cite style="font-style:italic">Das Problema bovimum des Archimedes</cite>. In: <cite style="font-style:italic">Zeitschrift für Mathematik und Physik: Historisch-literarische Abtheilung</cite>. Band 25. B. G. Teubner, Leipzig 1880, S. 121–136, 153–171 (<a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044102937786&view=1up&seq=612&size=125">hathitrust.org</a> [PDF; abgerufen am 6. Februar 2025]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Rinderproblem+des+Archimedes&rft.atitle=Das+Problema+bovimum+des+Archimedes&rft.au=Bernhard+Krumbiegel%2C+August+Amthor&rft.btitle=Zeitschrift+f%C3%BCr+Mathematik+und+Physik%3A+Historisch-literarische+Abtheilung&rft.date=1880&rft.genre=book&rft.pages=121-136%2C+153-171&rft.place=Leipzig&rft.pub=B.+G.+Teubner&rft.volume=25" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Johan Ludvig Heiberg (Hrsg.): Archimedis opera omnia cum commentariis Eutocii. E codice Florentino recensuit, latine uertit notisque illustrauit. <a rel="nofollow" class="external text" href="https://www.wilbourhall.org/pdfs/Archimedes_VOL_II.pdf">Bd. 2</a> (PDF; 13,3 MB). Teubner, Leipzig 1881, S. 446–455.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Georg Heinrich Ferdinand Nesselmann: Versuch einer kritischen Geschichte der Algebra. Bd. 1: Die Algebra der Griechen. Reimer, Berlin 1842. ND Minerva, Frankfurt 1969, S. <a rel="nofollow" class="external text" href="http://books.google.de/books?id=NCM1AAAAcAAJ&pg=PA482">482</a> (Protokoll), <a rel="nofollow" class="external text" href="http://books.google.de/books?id=NCM1AAAAcAAJ&pg=PA483">483</a> (Z. 1–30), <a rel="nofollow" class="external text" href="http://books.google.de/books?id=NCM1AAAAcAAJ&pg=PA486">486–487</a> (Z. 31–44).

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Hugh C. Williams, R. Angus German, C. Robert Zarnke: Solution of the cattle problem of Archimedes. In: Mathematics of Computation. 19, 1965, S. 671–674.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Merriman, Mansfield: The Cattle Problem of Archimedes. In: Popular Science Monthly. 67, 1905, S. 660–665.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Archimedes in the 21st Century: The Cattle Problem, Computer Output (2nd Part) <a rel="nofollow" class="external autonumber" href="https://www.math.nyu.edu/~crorres/Archimedes/Cattle/computer2/computer_output.html">[1]</a>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> H. W. Lenstra Jr.: All solutions to the cattle problem of Archimedes. S. 187 (<a rel="nofollow" class="external text" href="http://www.ams.org/notices/200202/fea-lenstra.pdf">PDF).</a>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Quellen">Quellen</h2></div>

<ul><li>G. Nesselmann: Die Algebra der Griechen. Reimer, Berlin 1842 (Nachdruck: Minerva Verlag, Frankfurt am Main 1969, ISBN 3-86598-328-6).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.cs.drexel.edu/~crorres/Archimedes/Cattle/Statement_greek.html">Seite mit dem griechischen Text, wie ihn Lessing in Wolfenbüttel entdeckt hat.</a></li>

<li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ArchimedesCattleProblem.html">Archimedes’ Cattle Problem.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li>

<li><a rel="nofollow" class="external text" href="https://www.math.nyu.edu/~crorres/Archimedes/Cattle/Statement.html">Englische Übersetzung von Ivor Thomas in Loeb Classics</a></li>

<li>Numberphile: <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=dNxyFtqcNss">The Archimedes Number</a> auf <a href="YouTube" title="YouTube">YouTube</a>, 24. November 2019, abgerufen am 24. Dezember 2019.</li></ul></div><div><div class="zim-footer">

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