  Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer
with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ is a sufficiently
small constant. Let $k(p,\,n)$ denote the least positive integer $k$ such that
for $x=-k,\,\dots,\,-1,\,1,\,2,\,\dots,\,k$, the numbers $x^n\pmod p$ yield all
the non-zero $n$-th power residues modulo $p$. In this paper, we shall prove $$
k(p,\,n)=O(p^{1-\delta}), $$ which improves a result of S. Chowla and H. London
in the case of large $n$.
