The factorial separation conjecture

Abstract

We propose the Factorial Separation Conjecture (FSC), a strict strengthening of earlier factorial gap conjectures: $|n!-m^k|>n$ for all integers $n\ge8$ and $m,k\ge2$. The inequality excludes every displacement up to and including the factorial index. We prove the conjecture unconditionally for exponents divisible by three, for even exponents at least four, and for every exponent at least two when the largest prime divisor of the base is at most $n+2$. For exponents divisible by five, we prove the finite interval $8\le n\le1100$ and all sufficiently large indices, with an effective but uncomputed threshold. The conclusion for every $n\ge8$ and every exponent divisible by five is proved under the Generalized Riemann Hypothesis (GRH) for a specified family of Dedekind zeta functions. Here GRH asserts that their nontrivial zeros have real part $1/2$. The arguments combine valuations, cyclotomic factorizations, cubic reciprocity, character estimates and explicit formulas. All finite comparisons are justified in the text. The general conjecture remains open.