The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

153

R. W. D. CHRISTIE, "Prove that the Pellian equation
n2 - PYn2 = 1 can be put into a trigonometrical form,
and give the general solution, thus for n = 3,
X3 = cos 30 = 4 cos3 0 - 3 cos 0 = 4x' - 3x,
Y3 = sin 30 = 4(cos2 0 - 1) sin 0 = 4x2y - y,"
Mathematical questions from the Educational Times,
vol. IX (2), p. 11l, London, 1906.
Let
Xn 2 py2 = 1
then
2n - 2n-3n 2 n-n(n - 3) -4
Y^ __Xn __ n-2 n-4
-0! 1! 2!
2-7n(n - 4)(n - 5) 6
3! x ~ '.
2n-2m-ln(n - m)! nm
-~ mm!(n-2m)!
2n-1 2n-3(n - 2)
y xn-ly 1! -3y
2n- (n - 3)(n - 4)
--! xn-a-y - * *.
2! -
2n-2m-l(n -m- 1)!xn2m
0 m!(n - 2m - 1)!
which coincides with X = cos nO, Y = sin nO, when cos nO
is developed from cosn 0, and sin nO is developed in the
form
sin 0 cosn 0 = sin 0 [(2 cos 0) n1 - (n - 2(2 cos 0) n(n - 3(n 4) (2 cos 0) ~.].
+ 2n3( (2 COS )n-5
A. H. BELL, J. BLAIKIE, F. N. MAYERS, "When is a
triangular number a pentagonal number?" Mathematical questions from the Educational Times, vol. IX (2),
p. 40, London, 1906.
This depends on the solution of the equation
x2 - 3y2 = 1.