The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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satisfies the equation) we can substitute 0 for n and d for
m in (1). Then again putting b = 0 and d = 1, he is in
a position to write down any number of successive solutions of
av2 + 1 = u2
when one solution,
v = q, u = p,
is known. The values of v are
O, q, 2pq, 4qp2 - q,..
and the corresponding values of u are
1, p, 2p2- 1, 4p3-3p, *,
the law of formation being that if A, B, are two consecutive values in either series, then the next following value
is 2pB - A.
But how is the fundamental solution, p, q, of the Pell
equation to be found? Euler first points out that when
a has one of many particular forms, the values, p, q,
which satisfy the equation, can at once be written down.
The simplest of these forms are found when a is one more
or one less than a perfect square. The following are the
cases mentioned by Euler:
a = e2- 1, q = 1, p = e;
a = e2 1, q = 2e, p = 2e2 +1;
a = a2e2b 2aeb-, q = e, p = aeb+l 1
(where a may even be fractional provided aeb-l is an
integer);
a = (aeb + 3eu)2 + 2aeb-1 + 23eu-1, q = e,
p = aeb+l + eu+l + 1;
a = -aCk2e2b aeb-l, q = ke,

p = 1 ak2eb+l - 1.