Hint for 8.21: Since we know c is less than 50 we actually have a decent bound for the values of m and n. For example, we know for sure m is at most 7. It's not too hard to list out from here. There are actually not too many triples.

Hint for 8.23: Start by making sure Example 8.3 makes sense (even just fully understanding the statement, the proof isn't needed for 8.23). Note this gives a restriction on what M can be. Hint for the rest: what are some the the smallest Pythagorean triples? With reference to Example 8.3 your answer with work can actually be very short for 8.23.