On Two Questions of Ordowski About the Largest Exponent in a Prime Factorization
Year: 2026
Type: preprint
Abstract: Let a(n) be the largest exponent in the prime factorization of n (OEIS A051903). In 2019 T. Ordowski asked whether there are odd n with a(n) > 1 such that (II) b^n = b^a(n) mod n for all b, or (III) 2^n = 2^a(n) mod n. (II) has no solutions. This was observed earlier by T. Adamczewski in reviews of the Formal Conjectures statements, and we give a lifting-the-exponent proof. We show that (III) has solutions. The prime attaining a(n) must be a Wieferich prime. For all Wieferich primes below 6.7*10^15 this forces a(n) = 2 and square part 3511^2, and the smallest solution is n = 7*631*881*3511^2*201961 = 9687963167864344937, established with the Wieferich search of Dorais and Klyve and two independent exhaustive scans. Both answers are formalized in Lean 4 with Mathlib against the statements OeisA51903.conjecture2 and conjecture3 of Google DeepMind's Formal Conjectures repository, using only the standard axioms. The repository contains the paper (LaTeX source and PDF), the Lean files, a reproducible check script, independent Python checks and all logs.
Authors: Mishra, Ansh, Senthilkumar, Aryan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214933922