View source for Archimedes's cattle problem

{{Short description|Mathematical problem in number theory}}
[[File:Archimedes_cattle_problem.svg|thumb|300px|Smallest solution to Archimedes's cattle problem with each icon representing around {{10^|206543}} cattle]]
'''Archimedes's cattle problem''' (or the '''{{lang|la|problema bovinum}}''' or '''{{lang|la|problema Archimedis}}''') is a problem in [[Diophantine equation#Diophantine analysis|Diophantine analysis]], the study of [[polynomial equation]]s with [[integer]] solutions. Attributed to [[Archimedes]], the problem involves computing the number of cattle in a herd of [[Helios|the sun god]] from a given set of restrictions. The problem was discovered by [[Gotthold Ephraim Lessing]] in a Greek manuscript containing a poem of forty-four lines, in the [[Herzog August Library]] in [[Wolfenbüttel]], [[Germany]] in 1773.<ref>{{cite book |last=Lessing |first=Gotthold Ephraim |date=1773 |title=Zur Geschichte und Litteratur: aus den Schätzen der Herzoglichen Bibliothek zu Wolfenbüttel, Zweyter Beytrag |language=German, Greek |trans-title=On History and Literature: from the treasures of the ducal library at Wolfenbüttel, second article |publisher=Fürstlicher Waysenhaus |location=Braunschweig, (Germany) |pages=421–425 |url=http://digitale.bibliothek.uni-halle.de/vd18/content/pageview/4023090 |archive-date=2018-09-17 |access-date=2018-09-17 |archive-url=https://web.archive.org/web/20180917182145/http://digitale.bibliothek.uni-halle.de/vd18/content/pageview/4023090 |url-status=dead }} From pp. 422–423: "''{{lang|de|Denn, wie gesagt, das Problem soll, wenn es nicht von dem Archimedes selbst abgefaßt worden, doch von ihm für werth erkannt seyn, daß er es den Eratosthenes geschicket hätte, um es den Meßkünstern zu Alexandria zur Auflösung vorzulegen. Dieses besagt die Aufschrift;{{nbs}}...}}''" (For, as said [above], the problem [Greek: ΠΡΟΒΛΗΜΑ] shall, if it had not been composed by Archimedes [Greek: Α'ΡΧΙΜΗΔΗΣ] himself, yet have been recognized by him [as so] worthy that he would have sent it to Eratosthenes [Greek: ΕΡΑΤΟΣΘΕΝΗΝ], in order to submit it to the surveyor at Alexandria for solution. The title says this;{{nbs}}...) See pages 423–424 (in Greek).</ref>

The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by {{ill|lt=Carl Ernst August Amthor|August Amthor|de}} (1845–1916), headmaster of the ''{{lang|de|Gymnasium zum Heiligen Kreuz}}'' ([[Gymnasium (school)|Gymnasium]] of the Holy Cross) in Dresden, Germany.<ref>{{cite journal |last1=Krumbiegel |first1=B. |last2=Amthor |first2=A. |date=1880 |title=Das Problema bovinum des Archimedes |language=German, Greek, Latin |trans-title=The cattle problem of Archimedes |journal={{lang|de|Zeitschrift für Mathematik und Physik: Historisch-literarische Abtheilung}} [Journal for Mathematics and Physics: Historical-literary section] |volume=25 |pages=121–136, 153–171 |url=https://babel.hathitrust.org/cgi/pt?id=hvd.32044102937786&view=1up&seq=611&size=125}}</ref><ref>Biographical information about August Amthor:
*Amthor's full name appears in: {{cite book |author=(School administration) |date=1876 |title=Programm des Gymnasiums zum Heiligen Kreuz in Dresden |language=German |trans-title=Program of the Gymnasium of the Holy Cross in Dresden |publisher=K. Blochmann und Sohn |location=Dresden, Germany |page=31 |url=https://books.google.com/books?id=4HPCnlJnDxUC&pg=PA31}}
*Brief biography about Amthor appears in: {{cite book |editor1-last=Singer |editor1-first=Isadore |editor2-last=de Leon |editor2-first=Edward Warren |title=International Insurance Encyclopedia |date=1910 |publisher=American Encyclopedic Library Association |location=New York, New York, USA |volume=1 |page=18 |chapter-url=https://books.google.com/books?id=GNMqAAAAYAAJ&pg=PA18 |chapter=Amthor, August (Ph.D.)}}</ref><ref>The problem was solved independently in 1895 by Adam Henry Bell, a surveyor and civil engineer of Hillsboro, Illinois, USA. See:
*{{cite journal |last1=Bell |first1=A.H. |title=On the celebrated 'Cattle Problem' of Archimedes |journal=The Mathematical Magazine |date=1895 |volume=2 |pages=163–164}}
*{{cite journal |last1=Bell |first1=A.H. |title=The 'Cattle Problem' by Archimedes 251 B.C |journal=American Mathematical Monthly |date=1895 |volume=2 |pages=140–141 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.b3627622&view=1up&seq=154}}
*Bell's full name appears in: {{cite book |editor1-last=Bateman |editor1-first=Newton |editor2-last=Selby |editor2-first=Paul |title=Historical Encyclopedia of Illinois |date=1918 |publisher=Munsell Publishing Co. |location=Chicago, Illinois, USA |volume=2 |pages=1049–1050 |chapter-url=https://books.google.com/books?id=GygPAQAAIAAJ&pg=PA1050 |chapter=Fish, Albert E.}}; see p. 1050.
*Bell's occupations appear in: {{cite journal |last1=Merriman |first1=Mansfield |title=The cattle problem of Archimedes |journal=Popular Science Monthly |date=November 1905 |volume=67 |pages=660–665 |url=https://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_67/November_1905/The_Cattle_Problem_of_Archimedes}}; see p. 664.</ref> Using [[logarithmic table]]s, he calculated the first digits of the smallest solution, showing that it is about {{val|7.76e206544}} cattle, far more than could fit in the [[observable universe]].<ref name="Lenstra2002">{{Citation
 | last = Lenstra | first = H. W. Jr.
 | author-link = Hendrik Lenstra
 | year = 2002
 | title = Solving the Pell Equation
 | journal = [[Notices of the American Mathematical Society]]
 | volume = 49
 | issue = 2
 | pages = 182–192
 | mr = 1875156
 | url = https://www.ams.org/notices/200202/fea-lenstra.pdf
}}</ref> The decimal form is too long for humans to calculate exactly, but [[Arbitrary-precision arithmetic|multiple-precision arithmetic]] packages on computers can write it out explicitly.

==History==
In 1769, [[Gotthold Ephraim Lessing]] was appointed librarian of the Herzog August Library in [[Wolfenbüttel]], Germany, which contained many Greek and Latin manuscripts.<ref name="rorres">{{cite web |url=http://www.mcs.drexel.edu/~crorres/Archimedes/Cattle/Statement.html |title=Archimedes' Cattle Problem (Statement) |access-date=2007-01-24 |last=Rorres |first=Chris |archive-url=https://web.archive.org/web/20070124203443/http://www.mcs.drexel.edu/~crorres/Archimedes/Cattle/Statement.html |archive-date=24 January 2007  }}</ref> A few years later, Lessing published translations of some of the manuscripts with commentaries. Among them was a Greek poem of forty-four lines, containing an arithmetical problem which asks the reader to find the number of cattle in [[The Cattle of Helios|the herd of the god of the sun]]. It is now generally credited to Archimedes.<ref>{{cite book |last=Fraser |first=P.M. |author-link=Peter Fraser (classicist)|year=1972 |title=Ptolemaic Alexandria |publisher=[[Oxford University Press]]}}</ref><ref>{{cite book |last=Weil |first=A. |author-link=André Weil |year=1972 |title=Number Theory, an Approach Through History |publisher=[[Birkhäuser]]}}</ref>

==Problem==
The problem, as translated into English by Ivor Thomas, states:<ref>{{cite web |title=The Cattle Problem Statement (English) |url=https://www.math.nyu.edu/~crorres/Archimedes/Cattle/Statement.html |website=New York University |access-date=11 July 2022}}</ref>
<blockquote>
If thou art diligent and wise, O stranger, compute the number of [[Cattle of Helios|cattle of the Sun]], who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.

But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.
</blockquote>

==Solution==
The first part of the problem can be solved readily by setting up a [[System of linear equations|system of equations]]. If the number of white, black, dappled, and yellow bulls are written as <math>W, B, D,</math> and <math>Y</math>, and the number of white, black, dappled, and yellow cows are written as <math>w,b,d,</math> and <math>y</math>, the problem is simply to find a solution to

:<math>\begin{align}
 W &= \frac{5}{6} B + Y, \\
 B &= \frac{9}{20} D + Y, \\
 D &= \frac{13}{42} W + Y, \\
 w &= \frac{7}{12} (B + b), \\
 b &= \frac{9}{20} (D + d), \\
 d &= \frac{11}{30} (Y + y), \\
 y &= \frac{13}{42} (W + w),
\end{align}</math>

which is a system of seven equations with eight unknowns. It is [[Indeterminate system|indeterminate]] and has infinitely many solutions. The least positive integers satisfying the seven equations are

:<math>\begin{align}
 B &= 7\,460\,514 = 4657 \times 1602,\\
 W &= 10\,366\,482 = 4657 \times 2226,\\
 D &= 7\,358\,060 = 4657 \times 1580,\\
 Y &= 4\,149\,387 = 4657 \times 891,\\
 b &= 4\,893\,246, \\
 w &= 7\,206\,360, \\
 d &= 3\,515\,820, \\
 y &= 5\,439\,213,
\end{align}</math>

which is a total of {{val|50,389,082}} cattle,<ref name="merriman1905">{{cite journal |last1=Merriman |first1=Mansfield |title=The cattle problem of Archimedes |journal=Popular Science Monthly |date=November 1905 |volume=67 |pages=660–665 |url=https://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_67/November_1905/The_Cattle_Problem_of_Archimedes}}</ref> and the other solutions are integral multiples of these. Note that given the [[prime number]] ''p'' = 4657 then the first four numbers are multiples of ''p'', and both ''p'' and ''p+1'' will appear repeatedly below.

The second part of the problem states that <math>W+B</math> is a [[square number]], and <math>Y+D</math> is a [[triangular number]]. The general solution to this part of the problem was first found by A.&nbsp;Amthor<ref>B. Krumbiegel, A. Amthor, ''Das Problema Bovinum des Archimedes'', Historisch-literarische Abteilung der Zeitschrift für Mathematik und Physik 25 (1880) 121–136, 153–171.</ref> in 1880. The following version of it was described by [[H.&nbsp;W. Lenstra]],<ref name="Lenstra2002"/> based on [[Pell's equation]]: the solution given above for the first part of the problem should be multiplied by

:<math>n = \frac{(w^{4658j} - w^{-4658j})^2}{(4657)(79072)},</math>

where ''j'' is any positive integer and

:<math>w = 300\,426\,607\,914\,281\,713\,365 \sqrt{609} + 84\,129\,507\,677\,858\,393\,258 \sqrt{7766},</math>

Equivalently, squaring ''w'' results in

:<math>w^2 = u + v\sqrt{(609)(7766)},</math>

where <math>(u, v)</math> is the fundamental solution of the [[Pell equation]]

:<math>u^2 - (609)(7766) v^2 = 1.</math>

The size of the smallest herd that could satisfy both the first and second parts of the problem is then given by ''j'' = 1 and is about <math>7.76 \times 10^{206\,544}</math> (first solved by Amthor). Modern computers can easily print out all digits of the answer. This was first done at the [[University of Waterloo]], in 1965 by [[Hugh C.&nbsp;Williams]], R.&nbsp;A. German, and Charles Robert Zarnke. They used a combination of the [[IBM&nbsp;7040]] and [[IBM&nbsp;1620]] computers.<ref>{{cite web |title=Unbundling Computing at The University of Waterloo |author=Harold Alkema and Kenneth McLaughlin |publisher=[[University of Waterloo]] |year=2007 |url=http://www.cs.uwaterloo.ca/40th/Chronology/printable.shtml |access-date=April 5, 2011 | archive-url=https://web.archive.org/web/20110404172741/http://www.cs.uwaterloo.ca/40th/Chronology/printable.shtml| archive-date=4 April 2011 | url-status=live}} (includes pictures)</ref>

==Pell equation==
The constraints of the second part of the problem are straightforward and the actual [[Pell equation]] that needs to be solved can easily be given. First, it one asks that ''B'' + ''W'' should be a [[Square number|square]], or using the values given above,

:<math>B + W = 7\,460\,514\,k + 10\,366\,482\,k = (2^2)(3)(11)(29)(4657)k,</math>

thus one should set ''k'' = (3)(11)(29)(4657)''q''<sup>2</sup> for some integer ''q''. That solves the first condition. For the second, it requires that ''D'' + ''Y'' should be a [[triangular number]]:

:<math>D + Y = \frac{t^2 + t}{2}.</math>

Solving for ''t'',

:<math>t = \frac{-1 \pm \sqrt{1 + 8(D + Y)}}{2}.</math>

Substituting the value of ''D'' + ''Y'' and ''k'' and finding a value of ''q''<sup>2</sup> such that the [[discriminant]] of this quadratic is a perfect square ''p''<sup>2</sup> entails solving the [[Pell equation]]

:<math>p^2 - (2^2)(609)(7766)(4657^2) q^2 = 1.</math>

Amthor's approach discussed in the previous section was essentially to find the smallest <math>v</math> such that it is integrally divisible by <math>2\times 4657</math>. The fundamental solution of this equation has more than 100,000 decimal digits.

==References==
{{Reflist}}

==Further reading==
*{{citation|title=The "Cattle Problem." By Archimedies 251 B. C.
|first=A. H. |last=Bell
|journal=The American Mathematical Monthly
|volume=2 |issue=5 |year=1895 |pages=140–141
|publisher=Mathematical Association of America
|doi=10.2307/2968125
|jstor=2968125}}
*{{cite book |last=Dörrie |first=Heinrich |title=100 Great Problems of Elementary Mathematics |publisher=[[Dover Publications]] |year=1965 |pages=3–7 |chapter=Archimedes' ''Problema Bovinum''}}
*{{cite journal |last=Williams |first=H. C. |author2=German, R. A.|author3=Zarnke, C. R. |title=Solution of the Cattle Problem of Archimedes |journal=Mathematics of Computation |year=1965 |volume=19 |pages=671–674 |doi=10.2307/2003954 |issue=92 |publisher=[[American Mathematical Society]] |jstor=2003954 |doi-access=free}}
*{{cite journal |doi=10.2307/2589706 |last=Vardi |first=I. |title=Archimedes' Cattle Problem |journal=American Mathematical Monthly |year=1998 |volume=105 |pages=305–319 | issue=4 |publisher=Mathematical Association of America|jstor=2589706 }}
*{{cite journal |last=Benson |first=G. |title=Archimedes the Poet: Generic Innovation and Mathematical Fantasy in the Cattle Problem |journal=Arethusa |year=2014 |volume=47 |pages=169–196 |issue=2 |publisher=[[Johns Hopkins University Press]] |doi=10.1353/are.2014.0008|s2cid=162393743 }}

==External links==
*{{OEIS el|1=A096151|2=Decimal expansion of the 206545-digit integer solution to Archimedes's cattle problem}}—Complete decimal solution to the second problem
*{{cite web |author1=Alex Bellos |author-link=Alex Bellos |title=Holy Cow that's a Big Number |url=https://www.youtube.com/watch?v=dNxyFtqcNss  |archive-url=https://ghostarchive.org/varchive/youtube/20211219/dNxyFtqcNss |archive-date=2021-12-19 |url-status=live|website=YouTube |date=24 November 2019 |publisher=[[Brady Haran]] |access-date=25 November 2019 |format=video}}{{cbignore}}
{{Archimedes}}

[[Category:Diophantine equations]]
[[Category:Large integers]]
[[Category:Works by Archimedes]]
[[Category:Cattle in literature]]