(a) $a^2+b^2=c^2\implies a^2=(c-b)(c+b)$. This means $a^2$ can be expressed in a way other than $a\cdot a$, meaning it has other prime factors meaning that $a$ is not prime. (b) Take $3-4-5$ triangle.

(a) No, since the sum of the squares of two odd primes must be even, hence the length of the hypotenuse cannot be prime if the other two lengths are odd primes. (b) Yes: (11, 60, 61).