The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

83

7, 3, 10, is 1. The equation corresponding to this form is
7 + 2.3o - 10w2 = 0,
from which
-3 = 79
= -10
and the integral part of o, say kl, is 1. For the next form,
(a', b', c'), make a' = - 10, the same as the last member of
the preceding form. Then b' is obtained by the congruence
b + b' 0 (mod- 10),
3 + b' O (mod- 10),
whence
b' -7 (mod - 10),
and whence
b' = 7.
Furthermore c' may be obtained from the equation
b'2 - a'c' = 79,
whence
49 - (- 10)c' = 79,
c' = 3.
Then the new form is (-10, 7, 3). In a similar manner the
remaining forms of the period are obtained; so that the entire period is (7, 3, -10), (-10, 7, 3), (3, 8, - 5), (- 5, 7, 6),
(6, 5, - 9), (- 9, 4, 7). Furthermore, the integers k2,
k3, k4, k5, k6, corresponding to kl of the first form are obtained at a glance. For example, add 8, the greatest
integer in 4'79, to 7 and divide by 3, and the integer is 5.
The k's are then 1, 5, 3, 2, 1, 1. This means that
3+ 79 1
1 +
5+
10 5 + - - 1
3+ 1
1
2+ 1
1+1+...
=(1,5,3,2,1, 1,...).