Let $a,b,c$ be integers, no sign restriction. Let $p$ be a given prime. How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ?
Note, from Heron's formula, $$ 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4 = (a + b + c) ( b + c - a) ( c + a - b) (a + b - c). $$
Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of primes of type prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ below $w$. How does the function $f(w)$ behave ? How fast does it grow ? Are those primes of type $A$ $mod$ $B$ for some integers $A$ and $B$ ? How to deal with this ?
Can this be solved without computing the class number ?