The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

19

is a known solution, put
pi = ap + -q, q1 = yp + q,
and it is sufficient to know the three simplest solutions in
order to find a, 3, y, 5; for, substituting the values of
p, q, Pi, ql, p2, q2, where p2, q2, are formed from pi, qi, by
the same law as pi, q1, are derived from p, q, we have four
simultaneous equations in four unknowns. Taking the
particular equation
x2 - 3y2 = 1
we easily find the first three solutions 1, 0; 2, 1; 7, 4.
Whence
2 = a, 1 = y, 7 = 2a +, 4 = 2y + 5,
and
a = 2, Y = 3, = 1, 6 = 2,
so that
pi =2p + 3q, q = p + 2q,
as already stated.
The celebrated Cattle Problem of Archimedes must
also be mentioned. Tannery1 says that instead of the
name of Pell in connection with the equation
x2 - Ay2 = 1,
the name of Archimedes ought to be joined to that of
Fermat, for the Cattle Problem shows that Archimedes
conceived the problem of the Pell equation in all its
generality. In this problem it is required to find the
number of bulls and cows of each of four colors, or to find
eight unknowns. The problem has a natural conclusion
at the close of seven conditions. Then, evidently by a
later hand, two other conditions are added; and the
language of the eighth is so ambiguous that it is doubtful
whether the sum of the number of bulls of two colors is
a square number or merely a number with two factors
nearly equal. But if all nine conditions are taken and
the eighth is made to refer to a square number, then the
1 P. Tannery, "Du role de la musique...," p. 175.