The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

51

is
1 5 29 169
1 x x 5 x 35 x 204
that is the values of y which make 8y2 + 1 a perfect
square are
1 1 5 1 5 29
1, 1 X 55, 1 X5 X 5 I X 5iX 56 X 535,
1 5 29 169
1 X 5 X 5 X 5 5 X 524, *
This product is evidently found through induction. The
first two factors are found by experiment as integers satisfying the given equation. The remaining parts may be
obtained according to the following rule: The numerator
of each fraction is equal to the denominator diminished
by the denominator of the preceding fraction, and each
denominator is equal to the numerator of the improper
fraction formed from the preceding mixed number.
Lord Brouncker1 subsequently found that the smallest
value of y was sufficient for determining all the others. If
ay12 + 1 = x12
then a(2x1y1)2 + 1 is also a square. Indeed,
a(2xlyl)2 + 1 = 4ayl2(ayl2 - 1) + 1 = (2ay12 - 1)2.
From this he deduced other values of y, as follows: If
Y = Yi, x = xi,
is the fundamental solution of
ay2 + 1 = x2,
and if z1 is set for 2x1 then the values of y are
Yl, ylZ, y l(z2 - 1), yl(zil - 2Z1), yl(Z14 - 312 + 1),
y1(z15 - 4z13 + 3z1), y1(z16 - 5z14 + 6z12 - 1),..,
"Commercium epistolicum," XVII, p. 789. "Oeuvres de Fermat,"
vol. III, p. 474.