PROVED
This has been solved in the affirmative.
Let $k\geq 2$ and $n_k(p)$ denote the least $k$th power nonresidue of $p$. Is it true that\[\sum_{p<x} n_k(p)\sim c_k \frac{x}{\log x}\]for some constant $c_k>0$?
Erdős
[Er61e] proved this when $k=2$, with\[c_2=\sum_{k=1}^\infty \frac{p_k}{2^k}.\]The general case was proved by Elliott
[El67b].
View the LaTeX source
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #980, https://www.erdosproblems.com/980, accessed 2026-01-16
