On the Diophantine Equation |ab^m − m!| = |ab^n − n!| and the Rigidity of Consecutive-Index Solutions

We investigate the Diophantine equation |ab^m − m!| = |ab^n − n!| for a nonzero integer parameter a and an integer base b ≥ 2. Using 2-adic valuations and Legendre's formula, we prove that every integer solution with 1 ≤ m < n and even b must have consecutive indices: n = m + 1. For the canonical case b = 2, we then determine necessary and sufficient conditions for the resulting parameter a to be an integer in both sign branches.