Archimedes's cattle problem

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. ==Problem==

The problem, as translated into English by Ivor Thomas, states: If thou art diligent and wise, O stranger, compute the number of 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. ==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 positive integers satisfying the seven equations are :\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} which is a total of 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 W+B is a square number, and Y+D is a triangular number. The general solution to this part of the problem was first found by A. Amthor ==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, or using the values given above, :B + W = 7\,460\,514\,k + 10\,366\,482\,k = (2^2 \times 3 \times 11 \times 29 \times 4657)k, thus one should set k = (3 \times 11 \times 29 \times 4657) q^2 for some integer q. That solves the first condition. For the second, it requires that D + Y should be a triangular number: :D + Y = \frac{t^2 + t}{2}. Solving for t, :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 :p^2 - (2^2 \times 609 \times 7766 \times 4657^2) q^2 = 1. Amthor's approach discussed in the previous section was essentially to find the smallest v such that it is integrally divisible by 2\times 4657. The fundamental solution of this equation has more than 100,000 decimal digits. ==References==