Can $b+c$ in the pythagorean triplets $(a, b, c)$ be a prime number? If $a^2 + b^2 = c^2$, can $b + c$ be a prime number?

Hint: $a^2+b^2=c^2\implies b^2-c^2=-a^2\implies (b+c)(b-c)=-a^2 \implies b+c=\frac{a^2}{c-b}$ $ \implies b+c=a^2 \times \frac{1}{c-b}$

