The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

87

and so on repeatedly. We may then find an infinity of
sets of values which satisfy (1).
II. Starting from
0 <x - y WA <,
and adding 2y /A, we have
0 < x +y A < + 2y'A.
Multiplying together the two relations just given, we have
0 < x2 - Ay2 < + 2 A/A < 1 + 2 IA.
Then if we designate by B some particular integer between
0 and 1 + 2 A/A, the equation
x2 - Ay2 = B
has an infinity of solutions.
III. Among the infinity of solutions of the equation
2 - Ay2 = B there can not be more than B2 sets of
values for x, y, such that, when x and y are divided by B,
the remainders constitute all the combinations of numbers
less than B. Therefore there are an infinity of sets which
give the same remainders, and we may write with xl, yl,
and x2, y2, two different sets of positive solutions,
Xi2 - Ay12 = X22- Ay22 = B,
x2 = xi+ aB, y2 = yi + fB;
and if we let
1 + oax - AOyl = x,
and
ayi - f3x = y,
then
(xl - y o/A)(x2 + Y2 /A) = B(x - y 'A),
and
(Xi + yi WA)(x2 - Y2 AA) = B(x + y WA).