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

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 $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$.