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 .
2026-03-30 23:57:03.1774915023
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How many natural solutions does the equation $x^2 - c y^2 = 1$ have?
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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.
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.