On the Burgess bound and Pythagorean triples involving primitive roots

Hai-Liang Wu; He-Xia Ni

On the Burgess bound and Pythagorean triples involving primitive roots

Number Theory 2026-08-05 v1 Combinatorics

Authors: Hai-Liang Wu , He-Xia Ni

Abstract

By combining the recent results of Pierce and Xu on the higher-dimensional Burgess bound with the classical one-dimensional Burgess bound, we prove that for any sufficiently large prime $p$ , there exists a Pythagorean triple $(a_p,b_p,c_p)$ with $a_p,b_p,c_p\in\mathbb{Z}\cap(0,p)$ such that $a_pb_p/2$ is a primitive root modulo $p$ . This confirms a conjecture of Z.-W. Sun for all sufficiently large primes.

Keywords

Burgess bound Pythagorean triples primitive roots higher-dimensional Burgess bound prime numbers

Cite

@article{arxiv.2608.04391,
  title  = {On the Burgess bound and Pythagorean triples involving primitive roots},
  author = {Hai-Liang Wu and He-Xia Ni},
  journal= {arXiv preprint arXiv:2608.04391},
  year   = {2026}
}