85 arithmetic/pell.p
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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.
85 arithmetic/pell.p
Find integer solutions to x^2 - 92y^2 = 1.
arithmetic/pell.s
x=1 y=0 x=1151 y=120 x=2649601 y=276240 etc.
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.
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