The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

3

In the Pell equation, and in such generalizations as
x2 -Ay2 = I 1, x2 -Ay2 = b, x2- Ay2 = a2,
it has been customary to restrict A to a positive nonsquare integer, and to seek integral solutions for x and y.
It is evident that no generality will be lost if the solutions
are further restricted to positive integers.
If it were permitted to make A < - 1, there would be
only two solutions in integers,
x= 1, y = 0.
For
A = -1,
only four solutions,
x= =- 1, y = O; x = 0, y=;
but for
A > 1, x2-Ay2= 1
has infinitely many solutions.
From every two solutions, xl, yi; x2, y2, the same or dif-,
ferent, a third can be obtained from the following identity,
(X + Y1i A) (x2 + Y2 wA)
xix2 + Ay1y2 + (Xly2 + X2y1) AA,
namely,
x = XlX2 + AY1y2, y = Xly2 + x2Y1.
All the solutions for which x > 0, y > 0, can be obtained
from the following formula in which xl, yi, is the fundamental solution, i.e., the solution in smallest integers,
excepting the solution 1, 0:
x + y = =(xi +y A)k (k = 1, 2, 3, * oc).
It is evident that the Pell equation is closely connected
with the primitive methods of approximating a square
root; indeed, that the latter are, in general, special cases
of the general equation. Among the first definite traces
that we have of these methods of approximation are those