Unified Formulas for Fermat Congruence Solutions via Cyclotomic Integers
Year: 2026
Type: preprint
Abstract: We present unified closed-form formulas for the number of solutions to x^n + y^n = z^n (mod p) for n = 3 (cubic) and n = 4 (quartic). For the cubic case, Sol_3(p) = p^2 + (p-1) B(a,b) where B(a,b) = 2b sigma(b) - (a-b) sigma(a-b) - (a+b) sigma(a+b) and sigma is the mod-3 sign function (Legendre symbol mod 3), verified for all 2,556 primes p congruent to 1 (mod 3) up to 50,000. For the quartic case, an analogous formula via Gaussian integers is verified for all 609 primes p congruent to 1 (mod 4) up to 10,000. Complete proofs are provided via Jacobi sums in the Eisenstein integers Z[omega], including the explicit six-case determination of primary associates. We prove the universal congruence B_n(p) congruent to n-2 (mod n) for every prime n, using a cancellation identity: the z=0 solution count and the Type 2 (diagonal) Jacobi sums cancel regardless of whether -1 is an n-th power. The same congruence is conjectured for all n and verified computationally through n = 512 with zero exceptions. We note that the Jacobi sum J(chi,chi) determining B(a,b) also equals the phase correlation function C(1) in p-level OAM quantum optical systems, connecting the Fermat solution count to measurement statistics in structured light experiments. The cubic formula further provides the arithmetic engine for the author's conformity gradient D(t) = W''/W on elliptic curve moduli (DOI: 10.5281/zenodo.19425788).
Author: Maino, Nicholas Clifford
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7150803365