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

Kübra Benli, Paul Pollack

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 $\epsilon >0$, and each…