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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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