If (a, b, c) is a primitive Pythagorean triple, the number c is called the hypotenuse of the triple, for reasons that should be self-explanatory. Every hypotenuse is nothing but a number of the form c = r² + s² where r, s are coprime and of different parity.

(a) Suppose c = s² + r² for some integers r, s. If p is a prime and p 3 mod 4 and p|c, explain why p|r and p|s.

(b) If a prime p divides the hypotenuse of primitive Pythagorean triple, show that p 1 mod 4. Hint: Rule out p = 2 and p 3 mod 4.

(c) If p is a prime and p 1 mod 4, explain why p is the hypotenuse of a primitive Pythagorean triple.