The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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functions of m, Xn, y, satisfy the differential equations
(M2 + 1)dmY + 3mdn (n2- l)y = 0,
(m2 + 1) dm2 + m - n2Xn = 0.
As another example, the equations
x2- (25m2 - 14m + 2)y2 = 1,
x2- (25m2 - 14m + 2)y2 = - 1,
with m a positive integer, have for solutions members
of the following series with even and odd index respectively,
yo = 0, Y1 = 5,. yn = 2(25m - 7)yn-1 + yn-2,
x0 = 1, x = 25m - 7, * X.n = 2(25m- 7)x,_1 + xn-2,
and we have
~d2yn dyn
(25m2-14m+2) dn +3(25m-7) d - 25(n2 - l)y 0 = 0,
d2x dx
(25m2-14mq-2) d - + (25m - 7) dmn 25n2x = 0.
J. ROMERO, "Equations de Pell," L'Intermediare des
mathematiciens, vol. IX, p. 182, Paris, 1902.
If x, y, is a solution of one of the equations
x2- Ay2 = = 1,
we have identically
(ny2 = x)2 - (n2y2 - 2nx + A)y2 = -- 1.
A. WEREBRUSOW, "Equations de Pell," L'Intermediare
des mathematiciens, vol. IX, p. 182, Paris, 1902.
In the equations x2 - Ay2 = - 1, A may have the
form a2m2 + 2bm t- c, provided the determinant
b2 - a2C = t 1.
ll~~~b