The Pell equation, by Edward Everett Whitford.

78

THE PELL EQUATION

whence
a'c' - D(a6' - y') (' - 6a') = b2,
and by subtracting
D(a5 - 3y)(a'a' - 1'y') = b2 - ac
from this equation, we have
(10) a'c' - D(ay' - ya')(f3' - 53') = ac,
and we also obtain
(11) 2b'c' - D(a'+ f3y'- 7/' - &6')(3' - 6/3') = 2bc,
(12) c'2 - D(03' - /3')2 = c2.
Let now the greatest common divisor of a, 2b, c be m,
and from this assumption determine numbers 2f, A3, (,
so that 3Ia + 23b + ~ c = m.1 Multiply equations (7),
(8), (9), (10), (11), (12) by 32, 2Nf8, $32, 22S, 23(, G2,
respectively, and for the sake of brevity let
(13) Xa' + 23b' + Cc' = T,
and
(14) 2(ac' - ya') + (aS' + y' - y' - 6')
+ G(~8' - a:') = U
where T, U, manifestly are integers. It is easily shown
that
T2 - DU2 = m2.
We are then led to the elegant conclusion that from any
two similar transformations which transform F into f,
there is found a solution in integers of the indeterminate
equation t2 - Du2 = m2, namely, t = T, u = U.
Gauss then goes on to show that from one transformation and one or more solutions of t2 - Du2 = m2, other
solutions may be found.
1 C. F. Gauss, op. cit., ~ 40.