A quadratic form of $p = 3k + 1$ primes

TL;DR

This paper proves that primes of the form 3k+1 or equal to 3 can be expressed as a quadratic form a^2+ab+b^2, using Zagier's proof approach.

Contribution

It provides a concise proof characterizing primes that can be written as a quadratic form based on their congruence class modulo 3.

Findings

Primes p=3 or p≡1 mod 3 can be expressed as a^2+ab+b^2.

The proof uses Zagier's one-sentence approach.

Characterization of primes via quadratic forms and modular arithmetic.

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 $p\equiv 1 \pmod{3}$.