The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

101

the extended table would lead to most erroneous results
in the fundamental solutions.
In a similar way when the period is odd, like the model
in the second extended table, only half the numbers in
third column need be computed. It has been the practice to use the formula
q2 = q 2 + q+l12
Thus
q2, = 222 + 252 = 484 + 625 = 1,109.
After p2, has been found by, say,
P2, = 4Aq,2 - 1 = 43,605,
or by using the formula
P2g = p,~+lq+l + pq., or P2, = kq2 + q2g-1,
we see that
x = 2p2,2 + 1 = 2(43,605)2 + 1 = 3,802,792,051,
and
y = 2p2, q2, = 2.43,605 1,109 = 96,715,890,
so that
43,6052 - 1,546.1,1092 = - 1
and
3,802,792,0512 - 1,546-9,671,5802= + 1.
If, however, in any problem we get p2 - Aq2 = 2,
then x and y are computed by the formulas x = p2 =F 1,
y = pq; and similar formulas are applied to p2 -Aq2= - 4.
Making use of the table of continued fractions which
is given in appendix A, the following fundamental solutions have been condensed from tables computed like
the two model extended tables just described. The
value of y is given before that of x. Where x2 - Ay2= -1
is solvable for example, for A = 1,514, the solution of
x2- Ay2 = - 1 is given first and then the solution of
x2- Ay2 = 1 directly after it, and the latter is further
distinguished by the fact that it consists of much larger
numbers.