<s>[INST] primes of the form $a^2+b^2=x^2-xy+y^2$? Let $a,b,x,y$ be strict positive integers.
Im intrested in primes $p$ such that $p=a^2+b^2=x^2-xy+y^2$.
What is the analogue PNT for these type of primes ? I think these primes are all the primes  $p \equiv 1 \pmod{12}$.
 [/INST] By Fermat's theorem on sums of two squares, $p$ can be written as $a^2+b^2$ iff $p \equiv 1 \pmod 4$ or $p = 2$. By this question, $p$ can be written as $x^2-xy+y^2$ iff $p \equiv 1 \pmod 3$ or $p = 3$. A prime $p$ satisfies both if and only if $p \equiv 1 \pmod {12}$, by the Chinese Remainder Theorem.
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