 Primes represented by x squared plus three y squared

Necessity follows by reducing modulo three. For a prime $p\equiv 1(\mathrm{mod}3)$, quadratic reciprocity gives $(-3/p)=1$, so the discriminant criterion for prime representation by a binary quadratic form constructs a primitive positive definite form of discriminant of a binary quadratic form $-12$ representing $p$. Reduction gives $|b|\le a\le c$ and $a\le 2$. The only primitive reduced form is $[1,0,3]$: the alternatives $[2,\pm 2,2]$ are not primitive. Integral equivalence preserves represented integers, establishing sufficiency.