The Pell equation, by Edward Everett Whitford.

94 THE PELL EQUATION
and the y's are the even numbers beginning with 4.
Another class contains
A = 7, 14, 23,.-,
x = 8, 15, 24,,
y = 3, 4, 5,.,
in which
A = m2 + 6m + 7,
x = m2 + 6m + 8 (m = 0, 1, 2, * ),
and the y's are the natural series beginning with 3.
Similar classes have been formed by Speckman1 for
y = 1 and y = 2. But classes can be formed for every
value of y. Thus for y = 5 we might have
A = 25m2 + (50 =- 8 =- 6)m + 26 - 8 =- (6 - 1),
x = 25m + 25 == 4 - 3,
y = 5,
in which m takes all integral values from 0 to o, and the
double signs may be read in any one of four ways, all plus,
all minus, alternately plus and minus beginning with
either sign.
For y any even number we have
A = m2n2 + (4m2 - 1)n + 4m2 = 2,
x = 2m2n + 4m2 = 1,
y = 2m;
and for y any integer, a general class would be
A = m2n2 + (2m2 = 2)n + m = 2,
x = m2n+m = 1,
= m,
in both of which classes m and n take all values from 0
to co.
1 Speckman, loc. cit., says that beyond y = 2 the formation of classes
seems to be very difficult.