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

Nagell proved that for each prime $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$, $p>7$, there is a prime $q<2p^{1\mathrm{/}2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $ϵ>0$, and each prime $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$ with $p > p_0(\epsilon)$,thenumberofprimecubicresidues$q < p^{1/2+\epsilon}$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 $p^{1\mathrm{/}9}$ prime quadratic residues $q<p$.