On the generalization of Golomb's conjecture

TL;DR

This paper investigates the distribution of primitive roots satisfying certain linear relations modulo large primes, providing an asymptotic formula that addresses an open problem, though the result was previously known.

Contribution

The paper offers an asymptotic formula for counting primitive roots with linear constraints, extending understanding of primitive root distributions.

Findings

Derived an asymptotic formula for primitive root counts

Addressed an open problem in primitive root distribution

Connected results to historical work by Carlitz (1956)

Abstract

Let $p$ be a sufficiently large prime number, $r$ be any given positive integer. Suppose that $a_1,\,\dots,\,a_r$ are pairwise distinct and not zero modulo $p$. Let $N(a_1,\,\dots,\,a_r;\,p)$ denote the number of $\alpha_1,\,\dots,\,\alpha_r,\,\beta$, which are primitive roots modulo $p$, such that $\alpha_1+\beta\equiv a_1,\,\dots,\,\alpha_r+\beta\equiv a_r\,({\rm mod}\,p).$ In the first version of this paper, we proved an asymptotic formula for $N(a_1,\,\dots,\,a_r;\,p)$ so that we could answer an open problem of Wenpeng Zhang and Tingting Wang. But we found that our result had been included in a paper of L. Carlitz in 1956, which is explained in the additional remark below.