= Primes represented by x squared plus three y squared
{title2=$p=x^2+3y^2\iff p=3\text{ or }p\equiv1\pmod3$}
Necessity follows by reducing modulo three. For a <prime> $p\equiv1\pmod3$, <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 $|b|\leq a\leq c$ and $a\leq2$. The only primitive reduced form is $[1,0,3]$: the alternatives $[2,\pm2,2]$ are not primitive. Integral equivalence preserves represented integers, establishing sufficiency.
Back to article page
