Let us have prime $p$ such that $p \nmid \gcd(a,b,c)$ and $p \mid a+b+c$. For what primes is it then impossible for $p \mid a^2+b^2+c^2$ ?
One example of such a prime is $p=5$:
If $5 \mid a^2+b^2+c^2$ ; then $5 \mid abc$ as the quadratic residues modulo $5$ are only $1,0,-1$. WLOG, let $5\mid a$. Then, $5 \mid b+c$ and $5 \mid b^2+c^2$ which shows that $5 \mid bc$ and thus, $5 \mid b$ and $5\mid c$. Contradiction.
It is probable that $5$ might be the only prime that shows the above characteristics. Are there any other primes or an infinite set of primes that share these properties due to the structure of their quadratic residues?