CUBES IN FINITE FIELDS AND RELATED PERMUTATIONS

Abstract Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in {Z}$ primitive root modulo p . $0<a_1<\cdots <a_n<p$ all the cubic residues in interval $(0,p)$ Then clearly sequence $a_1 \bmod p,\, a_2 p,\ldots , a_n p$ is permutation of $g^3 p,\,g^6 g^{3n} We determine sign this permutation.