This article is from the Puzzles FAQ,
by Chris Cole email@example.com and Matthew Daly
firstname.lastname@example.org with numerous contributions by others.
Find integer solutions to x^2 - 92y^2 = 1.
Each successive solution is about 2300 times the previous
solution; they are every 8th partial fraction (x=numerator,
y=denominator) of the continued fraction for sqrt(92) =
[9, 1,1,2,4,2,1,1,18, 1,1,2,4,2,1,1,18, 1,1,2,4,2,1,1,18, ...]
Once you have the smallest positive solution (x1,y1) you
don't need to "search" for the rest. You can obtain the nth positive
solution (xn,yn) by the formula
(x1 + y1 sqrt(92))^n = xn + yn sqrt(92).
See Niven & Zuckerman's An Introduction to the Theory of Numbers.
Look in the index under Pell's equation.