Archimedes' cattle problem

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Archimedes' 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 A. Amthor, and the actual numbers of cattle were computed in 1965 by H. C. Williams, R. A. German, and C. R. Zarnke, to the effect that the smallest herd which satisfies the conditions of the problem would contain approximately $7.760271 \times 10^{206544}$ cattle.

1 History
2 Problem
3 Solution
4 References
5 Further reading

[edit] History

In 1769, Gotthold Ephraim Lessing was appointed librarian of the Herzog August Library in Wolfenbüttel, Germany, which contained many Greek and Latin manuscripts.^[1] 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. The name of Archimedes appears in the title of the poem, it being said that he sent it in a letter to Eratosthenes to be investigated by the mathematicians of Alexandria. The claim that Archimedes authored the poem is disputed, though, as no mention of the problem has been found in the writings of the Greek mathematicians.^[2]

[edit] Problem

The problem, from an abridgement of the German translations published by Nesselmann in 1842, and by Krumbiegel in 1880, states:

Compute, O friend, the number of the cattle of the sun which once grazed upon the plains of Sicily, divided according to color into four herds, one milk-white, one black, one dappled and one yellow. The number of bulls is greater than the number of cows, and the relations between them are as follows:

White bulls $=\left(\frac{1}{2} + \frac{1}{3}\right)$ black bulls + yellow bulls,

Black bulls $=\left(\frac{1}{4} + \frac{1}{5}\right)$ dappled bulls + yellow bulls,

Dappled bulls $=\left(\frac{1}{6} + \frac{1}{7}\right)$ white bulls + yellow bulls,

White cows $=\left(\frac{1}{3} + \frac{1}{4}\right)$ black herd,

Black cows $=\left(\frac{1}{4} + \frac{1}{5}\right)$ dappled herd,

Dappled cows $=\left(\frac{1}{5} + \frac{1}{6}\right)$ yellow herd,

Yellow cows $=\left(\frac{1}{6} + \frac{1}{7}\right)$ white herd.

If thou canst give, O friend, the number of each kind of bulls and cows, thou art no novice in numbers, yet can not be regarded as of high skill. Consider, however, the following additional relations between the bulls of the sun:

White bulls + black bulls = a square number,

Dappled bulls + yellow bulls = a triangular number.

If thou hast computed these also, O friend, and found the total number of cattle, then exult as a conqueror, for thou has proved thyself most skilled in numbers.^[2]

[edit] Solution

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 $W, B, D,$ and $Y$ , and the number of white, black, dappled, and yellow cows are written as $w, b, d,$ and $y$ , the problem is simply to find a solution to:

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

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

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

which is a total of 50,389,082 cattle.^[2]

The general solution to the second part of the problem was found by A. Amthor^[3] in 1880. Amthor did not compute the total number of cattle, but rather stated that the number would contain 206,545 digits, and computed the first four. Only the first three of these were correct, however; the fourth was incorrect due to the use of insufficiently precise logarithms. The following version of it was described by H. W. Lenstra ^[4], based on Pell's equation: the solution given above for the first part of the problem should be multiplied by

(w 4658 j - w - 4658 j) 2 / 368238304

where

$w = 300426607914281713365 \times \sqrt{609} + 84129507677858393258 \times \sqrt{7766}$

and j is any positive integer. The size of the smallest herd that could satisfy both the first and second parts of the problem is given by j=1, and is approximately $7.760271 \times 10^{206544}$ .^[5] The actual number was first computed in 1965 by H. C. Williams, R. A. German, and C. R. Zarnke. The calculations were carried out by computer and required 7 hours and 49 minutes of computing time. ^[5]

[edit] References

^ Rorres, Chris. Archimedes' Cattle Problem (Statement). Retrieved on January 24, 2007.
^ ^a ^b ^c Merriman, Mansfield (1905). "The Cattle Problem of Archimedes". Popular Science Monthly 67: 660-665.
^ B. Krumbiegel, A. Amthor, Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift Für Mathematik und Physik 25 (1880) 121-136, 153-171.
^ Lenstra, H. W. (2002). "Solving the Pell equation". Notices of the American Mathematical Society 29 (2): 182-192.
^ ^a ^b Eric W. Weisstein, Archimedes' Cattle Problem at MathWorld.

[edit] Further reading

Dörrie, Heinrich (1965). "Archimedes' Problema Bovinum", 100 Great Problems of Elementary Mathematics. Dover Publications, pp. 3-7.
Williams, H. C.; German, R. A.; and Zarnke, C. R. (1965). "Solution of the Cattle Problem of Archimedes". Mathematics of Computation 19: pp. 671-674.