The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

89

The roots of this equation are given by the expression
27ri
z/z
e P, in which e and r have their ordinary meaning and
m designates an integer of the series
1, 2,3,...p - 1.
Among these integers there are (p - 1)/2 quadratic
residues1 and also the same number of non-residues of p
which we taken in a certain order and designate respectively by
al, a2, *.. apl and bi, b2,.* bpi.
2 2
Then from the theory of Gauss we have the two equations
27Ti 27Tri 2r'i ~
al — a2- a(p-l)]2
Y+Z~a=p=2(x-e p (x-e px ). (x-e ),
(2). b 2T ri
b2 27ri,. 2-i,
Y-Z p=2(xe )(x-e *... (x-e )/2
the upper or lower signs being taken according as p is of
the form 4n + 1 or 4n + 3, and Y and Z being polynomials in x with integral coefficients. Multiplying these
equations,
(4) 4X = y2:= pZ2.
As the numbers al, a2, a(p_l)/2, without regard to their
order, are the remainders given by the squares 12, 22, *
[(p - 1)/2]2 when divided by p, the first of the equations
(2) can be replaced by
12 27ri 22 2w p12 2r
(4) Y+Z = p==2(x-e P)(x-e P )...(x-e 2 ).
Suppose first that p is of the form 4n + 1. Then let
x = 1 in (3) and (4), and designate by g and h the corresponding integral values of Y and Z. Then
(5) g2 - ph2 = 4p
and
ri 27 i 1 p-l\227ri
12. 22 2F. Gauss. 62 18 t
g+ hp = 2(1-e )(1-e p).. (1 - e2 / p).
1 C. F. Gauss, "Disquisitiones," ~ 357, p. 637, Leipzig, 1801.