Cubic residuals for primes congruent to 1 modulo 6

Develop and prove a theory of cubic residues for primes p ≡ 1 (mod 6), analogous to Legendre’s quadratic residues, including a precise definition, foundational properties, and implications for related Diophantine equations.

Background

Building on discussions of quadratic residues and connections to sums of cubes, the paper reports that de Casteljau proposed a conjectural extension: a notion of cubic residual for primes of the form 6k + 1. Establishing such a theory would mirror the classical framework of quadratic residues.

The paper notes exploratory work by de Casteljau on residues modulo several primes and indicates this conjecture as a natural step toward a residue theory for cubic forms and associated arithmetic questions.