Nagell proved that for each prime $p\equiv 1\pmod{ 3}$, $p > 7$, there is a prime $q<2p^{ 1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $\epsilon > 0$, and each prime $p\equiv 1\pmod{ 3}$ with $p > p_0(\epsilon)$, the number of prime cubic residues $q < p^{ 1/2+\epsilon}$ exceeds $p^{ \epsilon/30}$. Our argument, like Nagell's, is rooted in the law of cubic reciprocity; somewhat surprisingly, character sum estimates play no role. We use the same method to establish related results about prime quadratic and biquadratic residues. For example, for all large primes $p$, there are more than $p^{ 1/9}$ prime quadratic residues $q<p$.
Categories: math.NT