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How many natural solutions has equation $x^2-cy^2=1$ depends on value of $c$ . I think I've seen this problem somewhere as a theorem but I can't remember where .

Ali Caglayan
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Antony
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    May be here? http://en.wikipedia.org/wiki/Pell%27s_equation – gammatester Oct 22 '14 at 09:08
  • @gammatester , yes it is . Write it as answer and I shall accept it .Thank you! – Antony Oct 22 '14 at 09:10
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    This equation is also Project Euler's introduction to diophantine equations in [problem 66] (https://projecteuler.net/problem=66) or, as I like to call it, the Project Euler hazing ritual. – Mike Oct 22 '14 at 09:39

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The equation $X^2 - d Y^2 = 1$ for $d$ a positive integer, is called Pell equation (while Pell actually had not much to do with it); having this name at hand it will be easy to find lots of information on it. It indeed has infinitely many integral solutions.

One reason this equation is relevant is that it solutions are linked to invertible elements in the ring of algebraic integers of real quadratic fields.

quid
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  • I should also have said "except if $d$ is a perfect square" when saying the equation has infinitely many solutions. Good the other answer got expanded. – quid Oct 22 '14 at 10:45
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This Pell's equation, here is the entry at Wikipedia. Among others there you can find the result from Lagrange, that as long as c is not a perfect square, Pell's equation has infinitely many distinct integer solutions, and how to find all solutions once you have a fundamental solution.

gammatester
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  • Perhaps quoting some content from the wikipedia article on why the contextual equations has these solutions etc. rather than just giving a link. – Ali Caglayan Oct 22 '14 at 09:43