ON THE GENERALIZED BROCARD–RAMANUJAN EQUATIONS

We prove that the generalized Brocard-Ramanujan Diophantine
equation n!+A = mk has finitely many integer solutions for any fixed integers
A and k ≥ 2. The proof evaluates the intersection of an additive continuous
bound, derived from the Taylor series remainder of e, and a multiplicative
discrete bound imposed by congruences modulo a primorial. We parameterize
the surviving residue classes and apply Weyl differencing with the Erd˝os-Tur´an
inequality to bound the discrepancy of the corresponding fractional parts. The
discrepancy error decays at a rate dominating the target continuous interval,
establishing the asymptotic independence of the conditions. Summing the
expected local densities of valid candidates over the infinite domain yields a
convergent counting function bound, completing the proof.