The Pell equation, by Edward Everett Whitford.

36

THE PELL EQUATION

We apply the same series of operations to the equation
just written that were applied to the first equation, and
obtain
aq22 + s2 = P22,
and so on repeatedly until
Sr = 1, - 2, or 4.
If one solution of
aq2 + s = p2
is known, then a solution of
ay2 + s2 = x2 is X = p2 + aq2, y = 2pq;
and if
s = - 2,
2pq is even, and therefore
x2 = ay2 + 4
is divisible by 4, so that (p2 + aq2) and pq form an
integral solution of
ay2 + 1 = 2.
One way of handling the case
s= 4
has been mentioned on page 29.
The Hindus applied the cyclic method successfully to
all problems of this kind which they attempted, even to
those involving numbers of considerable magnitude. For
x2 - 61y2 = 1,
Bhaskara finds
y = 226,135,980, x = 1,766,319,049.
He does not avoid negative numbers but uses positive or
and since p and q are relatively prime (pq- - 1) is divisible by q, and
therefore, (pql - l)/q is an integer, pi; and from above, (r12 - a)/s is an
integer, sl, and the sum aq12 + sl is the square of an integer.