A quadratic form of $p = 3k + 1$ primes
Bat-Od Battseren, Bayarmagnai Gombodorj
TL;DR
This work characterizes primes representable as $p=a^2+ab+b^2$ by adapting Zagier's one-sentence proof technique. It leverages a fixed-point Lemma for group actions and a carefully constructed involution on the solution set of $3x^2+yz=p$ to force the existence of a fixed point yielding the desired representation. The authors first note the trivial case $p=3$ and then handle $p>3$ with $p\equiv1\pmod{3}$, using a partition into $S_1,\dots,S_{10}$ and corresponding transformations to ensure an odd fixed-point count, leading to $p=x^2+x(x+z)+(x+z)^2$ and thus $p=a^2+ab+b^2$ with $a=x$, $b=x+z$. The approach connects to the known equivalence $p\equiv1\pmod{3}$ iff $p=x^2+3y^2$ and provides a constructive path from modular conditions to a quadratic form representation.
Abstract
We use Zagier's one-sentence proof approach to show that a prime number $p$ admits a form $p=a^2+ab+b^2$ for some integers $a$ and $b$ if and only if $p=3$ or $p\equiv 1 \pmod{3}$.