On the Diophantine Equation |ab^m − m!| = |ab^n − n!| and the Rigidity of Consecutive-Index Solutions
Year: 2026
Type: preprint
Abstract: We study the Diophantine equation |ab^m − m!| = |ab^n − n!| in an integer a ≠ 0, an integer base b ≥ 2, and integers 1 ≤ m < n. Using q-adic valuations and Legendre's formula, we prove that every solution satisfies n − m ≤ q − 1, where q is the smallest prime factor of b; for even b the indices are therefore consecutive. Applying a valuation argument to the consecutive case, we show that consecutive indices occur only for b = 2. Hence b = 2 is the only even base with solutions, no base divisible by 3 has any, and for odd b every solution has gap at least 2. For b = 2 we give exact integrality criteria, in terms of the 2-adic valuation and binary digit sum of m, for both sign branches, and identify the smallest solutions. We also prove that the number N(2^n) of admissible indices equals the Fibonacci number F(n+1), so N(X) grows like X^(log2 φ), where φ is the golden ratio.
Author: Kumar, Aditya
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204567142
Cites:
Cited by:
Related to: