A note on the sum of cubic residues with even and odd index

Abstract

Let p be an odd prime and $\ell \in {\mathbb {N}}$. We prove that the sum of cubic residues of $p^\ell$ with even index equals the sum of cubic residues of $p^\ell$ with odd index if and only if $p \equiv 1 \pmod 4$. We also prove that for $2p^\ell$, $p \equiv 1 \pmod 4$ is sufficient but not necessary for the two sums to be equal. We also present a closed-form expression for the sum and study some properties.