TITLE: If $a,b,c,a^2+b^2+c^2$ are primes, then $a$ or $b$ or $c$ is equal to $3$
QUESTION [5 upvotes]: Given primes $a,b,c$ such that $a^2+b^2+c^2$ is prime, then $3\in\{a,b,c\}$.
Tested for $a,b,c<500$.

REPLY [11 votes]: If $p\neq3$, $p$ a prime, then $p^2\equiv1\pmod3$. So if $3\notin\{a,b,c\}$ then $3\mid a^2+b^2+c^2$.