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

Number Theory 2025-11-26 v1 Group Theory

Abstract

We use Zagier's one-sentence proof approach to show that a prime number pp admits a form p=a2+ab+b2p=a^2+ab+b^2 for some integers aa and bb if and only if p=3p=3 or p1(mod3)p\equiv 1 \pmod{3}.

Keywords

Cite

@article{arxiv.2511.19763,
  title  = {A quadratic form of $p = 3k + 1$ primes},
  author = {Bat-Od Battseren and Bayarmagnai Gombodorj},
  journal= {arXiv preprint arXiv:2511.19763},
  year   = {2025}
}