Archimedes's cattle problem
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.
[2][3][4] 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.
[5] 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.
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.
[7][8] The problem, as translated into English by Ivor Thomas, states:[9] 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.
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.
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.
The least positive integers satisfying the seven equations are which is a total of 50389082 cattle,[10] and the other solutions are integral multiples of these.
The following version of it was described by H. W. Lenstra,[5] based on Pell's equation: the solution given above for the first part of the problem should be multiplied by where j is any positive integer and Equivalently, squaring w results in where
is the fundamental solution of the Pell equation 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
[12] 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.
For the second, it requires that D + Y should be a triangular number: Solving for t, 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 Amthor's approach discussed in the previous section was essentially to find the smallest
