The Pell equation, by Edward Everett Whitford.

20

THE PELL EQUATION

Cattle Problem involves the solution of the Pell equation
x2 4,729,494y2 =.
But it is very doubtful whether the Cattle Problem was
proposed by Archimedes or even in his time.1 If he did
propose it he certainly could not solve it.2 Nesselmann3
sums up the evidence by saying that the problem is clearly
at an end after seven conditions have been given, the
concluding words being "he who solves the problem must
not be unskilled in numbers." Its language and versification are against its authenticity. The scholiasts' solution
does not, as it claims, satisfy the whole problem but only
the first part. The impossibility of solution with Greek
numerals and the very large numbers used show that
the author, or authors, could not have seen what the
effect of the many heterogeneous conditions would be.
There is nothing impossible in the supposition that
Archimedes was in the possession of a general method of
solving such equations when the numbers involved were
not too great for manipulation in the Greek numerical
notation.4 But the other conditions are such that we
are not allowed to take the fundamental solution of the
Pell equation but must take a solution in which y is
divisible by 9,314, with the result that larger numbers of
cattle are involved than could by any possibility stand
upon the plains of Thranika or the whole island of Sicily
for that matter, numbers beyond comprehension or
possibility of computation. For instance the number of
yellow bulls would be
639,034,648,230,902,865,008,559,676,183.......
635,296,026,300,
where the dots indicate the omission of 68,834 periods of
1J. Struve and K. L. Struve, "Altes griechisches Epigramm," Altona,
1821.
2 S. Gunther, loc. cit.
3 G. H. F. Nesselmann, loc. cit.
4T. L. Heath, "Diophantus of Alexandria," loc. cit.