The Pell equation, by Edward Everett Whitford.

156

THE PELL EQUATION

one having been found. What is the simplest multiplier
for x2 - 601y2 = 1?" Mathematical questions from the
Educational Times, vol. XI (2), p. 39, 54, London, 1907,
Educational Times, vol. LIX, p. 350, Aug. 1, 1906, p. 412,
Sept. 1, 1906, London.
If one solution, x, y, is known for the equation
2 - =
x2 py2 1,
then since
(2x2 _ 1)2 2)2 = 4x2(x2 _ py2) _ 4x2 + 1 = 1
we may use 2x as a multiplier. In general
xn+1 = 2x, - x_-1, Y n+-l = 2XnYU - yn-1_
For 601 the multiplier is 2 (38,902,815,462,492,318,420,311,478,049).
R. W. D. CHRISTIE, "If 2 - py2 = 1 and X2 - pY2= 1,
prove (X)2 + Y2 = (xy)2 + y2 for an infinity of integral values of x, y, X, Y," Mathematical questions from
the Educational Times, vol. XI (2), p. 96, London, 1907.
This is proved by A. Cunningham as well as the proposer.
x2 -1 X2 - 1
y2 =- P = y (Xz - 1)2 = (X2 - )y2.
A. CUNNINGHAM, "Let
En = 22'", Fn = E + 1, Gn = E- 1,
show that every Fn ( > 5) may be expressed algebraically
in the form
2Fn = t2 - GnU2
and obtain the expression. Discuss the possibility of
obtaining a second expression,
2F,, = t'2 - GnU2,
which together with the former shall be suitable for
factorizing F,." Mathematical questions from the Educational Times, vol. XII (2), p. 21, London, 1907.