Archimedes's cattle problem

Archimedes's cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of 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.

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 Carl Ernst August Amthor [de] (1845–1916), headmaster of the Gymnasium zum Heiligen Kreuz (Gymnasium of the Holy Cross) in Dresden, Germany. Using logarithmic tables, he calculated the first digits of the smallest solution, showing that it is about 7.76×10²⁰⁶⁵⁴⁴ cattle, far more than could fit in the observable universe. The decimal form is too long for humans to calculate exactly, but multiple-precision arithmetic packages on computers can write it out explicitly.

In 1769, Gotthold Ephraim Lessing was appointed librarian of the Herzog August Library in Wolfenbüttel, Germany, which contained many Greek and Latin manuscripts. 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 herd of the god of the sun. It is now generally credited to Archimedes.

The problem, as translated into English by Ivor Thomas, states:

The first part of the problem can be solved readily by setting up a system of equations. If the number of white, black, dappled, and yellow bulls are written as ${\displaystyle W,B,D,}$and ${\displaystyle Y}$, and the number of white, black, dappled, and yellow cows are written as ${\displaystyle w,b,d,}$and ${\displaystyle y}$, the problem is simply to find a solution to

${\displaystyle {\begin{aligned}W&={\frac {5}{6}}B+Y,\\B&={\frac {9}{20}}D+Y,\\D&={\frac {13}{42}}W+Y,\\w&={\frac {7}{12}}(B+b),\\b&={\frac {9}{20}}(D+d),\\d&={\frac {11}{30}}(Y+y),\\y&={\frac {13}{42}}(W+w),\end{aligned}}}$

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

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

which is a total of 50389082 cattle, 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 ${\displaystyle W+B}$is a square number, and ${\displaystyle Y+D}$is a triangular number. The general solution to this part of the problem was first found by A. Amthor in 1880. The following version of it was described by H. W. Lenstra, based on Pell's equation: the solution given above for the first part of the problem should be multiplied by

${\displaystyle n={\frac {(w^{4658j}-w^{-4658j})^{2}}{(4657)(79072)}},}$

where j is any positive integer and

${\displaystyle w=300\,426\,607\,914\,281\,713\,365{\sqrt {609}}+84\,129\,507\,677\,858\,393\,258{\sqrt {7766}},}$

Equivalently, squaring w results in

${\displaystyle w^{2}=u+v{\sqrt {(609)(7766)}},}$

where ${\displaystyle (u,v)}$is the fundamental solution of the Pell equation

${\displaystyle u^{2}-(609)(7766)v^{2}=1.}$

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 ${\displaystyle 7.76\times 10^{206\,544}}$(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. Williams, R. A. German, and Charles Robert Zarnke. They used a combination of the IBM 7040 and IBM 1620 computers.

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, or using the values given above,

${\displaystyle B+W=7\,460\,514\,k+10\,366\,482\,k=(2^{2})(3)(11)(29)(4657)k,}$

thus one should set k = (3)(11)(29)(4657)q² for some integer q. That solves the first condition. For the second, it requires that D + Y should be a triangular number:

${\displaystyle D+Y={\frac {t^{2}+t}{2}}.}$

Solving for t,

${\displaystyle t={\frac {-1\pm {\sqrt {1+8(D+Y)}}}{2}}.}$

Substituting the value of D + Y and k and finding a value of q² such that the discriminant of this quadratic is a perfect square p² entails solving the Pell equation

${\displaystyle p^{2}-(2^{2})(609)(7766)(4657^{2})q^{2}=1.}$

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

• Bell, A. H. (1895), "The "Cattle Problem." By Archimedies 251 B. C.", The American Mathematical Monthly, 2 (5), Mathematical Association of America: 140–141, doi:10.2307/2968125, JSTOR 2968125
• Dörrie, Heinrich (1965). "Archimedes' Problema Bovinum". 100 Great Problems of Elementary Mathematics. Dover Publications. pp. 3–7.
• Williams, H. C.; German, R. A.; Zarnke, C. R. (1965). "Solution of the Cattle Problem of Archimedes". Mathematics of Computation. 19 (92). American Mathematical Society: 671–674. doi:10.2307/2003954. JSTOR 2003954.
• Vardi, I. (1998). "Archimedes' Cattle Problem". American Mathematical Monthly. 105 (4). Mathematical Association of America: 305–319. doi:10.2307/2589706. JSTOR 2589706.
• Benson, G. (2014). "Archimedes the Poet: Generic Innovation and Mathematical Fantasy in the Cattle Problem". Arethusa. 47 (2). Johns Hopkins University Press: 169–196. doi:10.1353/are.2014.0008. S2CID 162393743.

• OEIS sequence A096151 (Decimal expansion of the 206545-digit integer solution to Archimedes's cattle problem)—Complete decimal solution to the second problem
• Alex Bellos (24 November 2019). "Holy Cow that's a Big Number" (video). YouTube. Brady Haran. Archived from the original on 2021-12-19. Retrieved 25 November 2019.