Cubes in finite fields and related permutations

Abstract

Let $p=3n+1$ be a prime with $n\in \mathbb{N}=\{0,1,\cdots \}$, and let $g\in \mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots <a_n<p$ be all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the sequence $$a_1\mathrm{mod}p,a_2\mathrm{mod}p,\cdots ,a_n\mathrm{mod}p$$ is a permutation $s_p(g)$ of the sequence $$g^3\mathrm{mod}p,g^6\mathrm{mod}p,\cdots ,g^{3n}\mathrm{mod}p.$$ In this paper, we shall determine the sign of this permutation.

9 pages