If $a,b,c$ are positive integers and $a^2+b^2=c^2$ and $a$ is a prime, what can we conclude about primeness of b and c? Let $a,b,c$ be positive integers and they satisfy $a^2+b^2=c^2$, and if $a$ is prime, can we conclude whether $b$ and $c$, are both prime, composite or neither? If yes, why, if not why not?
I can conclude that $b$ and $c$ have to be one odd and the other one even using $a^2 = c^2-b^2=(c-b)(c+b)$. But I couldn't conclude anything about their primeness. Can anyone show me some ideas or its reasoning and maybe any useful and related theorems?
Thanks very much!

$c$ is prime and $b$ isn't :
$$3^2+4^2=5^2$$ 
both are composite :
$$7^2+24^2=25^2$$
However, $b$ can never be a prime. Proof : 
$a=2$ or $b=2$ is impossible (first pythagorean triplet is $3,4,5$)
if both $a$ and $b$ are odd prime numbers, then $a^2\equiv b^2\equiv 1 \mod 4$ (this is true for all odd numbers), and therefore $c^2\equiv 2\mod 4$ (impossible)
