On the Burgess bound and Pythagorean triples involving primitive roots

Hai-Liang Wu; He-Xia Ni

paper

On the Burgess bound and Pythagorean triples involving primitive roots

arXiv:2608.04391

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 $a_{p}, b_{p}, c_{p} \in Z \cap (0, p)$ such that $a_{p} b_{p} /2$ is a primitive root modulo $p$ . This confirms a conjecture of Z.-W. Sun for all sufficiently large primes.