We present the GADS-MCL Modular Step Theorem, a deterministic constraint on Fermat's difference-of-squares parameter derived from the multiplicative structure of the primorial ring Z/42Z. For any semiprime N = p·q with gcd(N, 42) = 1, the Fermat half-difference t = (p−q)/2 satisfies a fixed congruence modulo 21 determined solely by the residue class of N modulo 42. This reduces the effective search space of Fermat's method by a theoretical factor of 21, converting the standard step-1 iteration over t into a step-21 iteration. Empirical benchmarks over 4,446 test cases confirm speedups averaging 3.8x, reaching 15x for highly asymmetric factor pairs. A hybrid algorithm is proposed that combines standard Fermat (for nearly equal factors) with the modular-step method (for asymmetric cases), achieving robust performance across all configurations. The result is algebraically exact, requires no heuristics, and constitutes a natural extension of the GADS-MCL framework (Diaz Stefani, 2025–2026) to the classical Fermat method.
Mariano Francisco Diaz Stefani (Tue,) studied this question.