The Pell equation, by Edward Everett Whitford.

86

THE PELL EQUATION

based on Dedekind's modification of Dirichlet'sl demonstration.
I. Let y be an integer which varies from 0 to n so that
0 < y < n,
and let A be a positive non-square integer and x the integer
immediately superior to y IA, then
0 < x - yAIA < 1.
(The equality symbol is needed only for the case y = 0,
x = 1.) The value of any x - y lA is comprised between two of the fractions
0 1 2 n
n n n n
As there are only n intervals and as y, and therefore
x - y /A, may take n + 1 values, there is at least one
interval containing two values, and we may therefore
write
0 < (Xi - yi AA) - (X2 -y2 AA) <
or
0 < (xl - X2) - (Y1 - y2) 4A I <
Now I y - y2 is one of the values of y, and letting
xI - x2 be represented by x, we have
0 <x-y JA < < y.
n y
Now let m be an integer so large that the smallest value
of x - y AIA shall be greater than 1/m. Then proceeding
in the same manner, using m instead of n we get a new
set, x, y, which gives a new solution to
(1) 0 <x -y A <,
Dirichlet, "Zahlentheorie," 4th ed., p. 372. A. Aubry, "Theorie de
l'equation de Pell," Mathesis, vol. V (3), p. 233, Paris, 1905.