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!

$5^2+12^2=13^2$ and $7^2+24^2=25^2$, so I don't think there is anything interesting about primality of $b$ and $c$...
What you may say is that, except for trivial cases, $c+b\neq c-b$, hence we must have $c-b=1$ and $c+b=2b+1=a^2$, so $b=\frac{a^2-1}2$ and $c=\frac{a^2+1}2$. Hence, you find that exists exactly one such triple $(a,b,c)$ for every odd prime number.
