  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.
