On the generalization of Golomb's conjecture

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 ${\alpha }_1+\beta \equiv a_1,\dots ,{\alpha }_r+\beta \equiv a_r(\mathrm{mod}p).$ 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.