TITLE: Representations of some primes as $3x^2-4y^2$?
QUESTION [0 upvotes]: I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings:
$$
p\equiv11\pmod{12}\longrightarrow p=3x^2-4y^2
$$
Any help appreciated.

REPLY [1 votes]: Using quadratic number fields the basic idea is this: primes $p \equiv \pm 1 \bmod 12$ split in $K = {\mathbb Q}(\sqrt{3})$. Since $K$ has class number $1$, the prime ideals above $p$ are principal, hence $x^2 - 3y^2 = \pm p$. Now clearly $p \equiv 1 \bmod 12$ implies that the plus sign holds, whereas for $p \equiv 11 \bmod 12$ the minus sign must hold: $x^2 - 3y^2 = -p$. Looking at this equation modulo $4$ will tell you that $x$ is even.
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