Diophantine equations

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Consider Pell's equation $x^{2} - 8\,y^{2} = 1$. Its fundamental (smallest non-trivial) solution has $y = 1$. Find the corresponding positive integer $x$.

2. Warm-up

   Consider Pell's equation $x^{2} - 35\,y^{2} = 1$. Its fundamental (smallest non-trivial) solution has $y = 1$. Find the corresponding positive integer $x$.

3. Warm-up

   Consider Pell's equation $x^{2} - 15\,y^{2} = 1$. Its fundamental (smallest non-trivial) solution has $y = 1$. Find the corresponding positive integer $x$.

Problems shown under their original terms: Generated (template, solver-computed).