For 8.23 start by making sure Example 8.3 makes sense. Note this gives a restriction on what M can be. Hint for the rest: what are some the the smallest Pythagorean triples?

For 8.25, a natural first step might be to try (mod 2), so you get an equation just in y. However, this doesn't quite work (although we do know that y must be even). What if you try looking (mod 4) or (mod 8)?