Competition problem — Number theory

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Number theory Difficulty 6.0 National olympiad Prove it

12. Prove: If $x^{2}-d y^{2}=c$ has one solution, then it must have infinitely many solutions, where $d>1$ is a non-square number, and $c$ is an integer.

Solution

12. Let $x_{1}+y_{1} \sqrt{d}$ be a positive solution of $x^{2}-d y^{2}=1$ , and $u_{1}, v_{1}$ be solutions of the original indeterminate equation. Then $\left(x_{1}+y_{1} \sqrt{d}\right)^{n}(u+v \sqrt{d})=\left(u_{n}+v_{n} \sqrt{d}\right)(n=1,2, \cdots)$ gives $u_{n}, v_{n}$ which are all solutions of the original equation.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.