Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach

Abstract

Nagell proved that for each prime $p\equiv 1(\mathrm{mod}3)$, $p>7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $\hat{I}\mu >0$, and each prime $p\equiv 1(\mathrm{mod}3)$ with $p>p_0(\hat{I}\mu )$, the number of prime cubic residues $q<p^{1/2+\hat{I}\mu }$ exceeds $p^{\hat{I}\mu /30}$. 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 $q<p$.

7 pages