The Pell equation, by Edward Everett Whitford.

144

THE PELL EQUATION

{8[n3 + (n + 1)3] + 1}2 - [(2n + 1)2
+ 4]{4[n3 + (n + 1)3][n2 + (n + 1)2]}2 = 1,
give solutions besides 1, 0; for the equation x2 - Ay2 = 1,
where A has one of the forms
n2 1, n(k2n 2), 4n2 + 25n + 39, (2n + 1)2 + 4.
The author asks whether there is an analogous identity
for every value of A not a perfect square.
H. BROCARD, On a law of recurrence characteristic
of all the solutions of the Pell equation, L'Intermediare
des mathematiciens, vol. VIII, p. 59, Paris, 1901.
G. RICALDE, E. B. ESCOTT, "La loi de recurrence des
solutions de l'equation de Pell," L'Intermediaire des
mathematiciens, vol. VIII, p. 286, Paris, 1901.
A. BOUTIN, "Resolution complete de l'equation
x2- (Am2 + Bm + C)y2= 1 ouf A, B, C, sont des
entiers, par une infinite des polyn6mes en m," L'Intermediare des mathematiciens, vol. IX, p. 60, Paris, 1902.
The equations x2 - Ay2 = - 1, when we have A a
conveniently chosen function of the second degree of the
parameter m, are solved completely by an infinity of
polynomials in m. These polynomials satisfy certain
differential equations of the second order. For example,
if we have the equations
2 - (m2 + )y2 = 1, x2 (m2 + )y2 = 1,
the recurring series
Yo =, y1 = 1,. * y, = 2myn-i + yn-2,
Xo = 1, X1 = m,..* * n = 2mxn-l + Xn-2
(n = 2,3,...),
with x, y, of even index give solutions of the first equation
and x, y, of odd index to the second equation; and the