The Pell equation, by Edward Everett Whitford.

THE PELI EQUATION

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Then he finds X and Y, called first by him the fundamental solution, such that
X = EkA-,_ + 10-2, Y = kl,
in which the k's are obtained from the following formulas,'
k = 1,
kl = Xlk,
Z 2 = X2kl + k,
k = X3k2 + kl,
k4 = X4k3 -+ k2,
k5 = X5k4 + k3,
This solution, X, Y, satisfies the equation
X2 - BY2 = i 1.
In general,
(X + Y rB)n + (X - Y B)n
r= 2
2 B
in which n must be so taken that nu is even or odd according as r2 - Bs2 = + 1or - 1. When r2 - Bs2 = 1,
ngu must be even. If,u is even we may take any positive
integer for n; if,u is odd we may choose only even numbers
for n. Accordingly every equation of the form
r2 - Bs2 = 1
is solvable in integers.
For the equation r2 - Bs2 - 1 to be possible Lagrange stated the theorem that B could have no prime
factors except those of the form 4n + 1. This condition
is necessary, for otherwise B could not be a divisor of
x2 + 1; but is not sufficient, since, for example, there is
1 Op. cit., p. 64.