HEXAGONbeta

Joint distribution of factorial units and valuations in translated intervals

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

For every fixed finite set of primes, the corresponding prime valuations of factorials, reduced modulo arbitrary fixed integers, and the prime-free parts, reduced modulo arbitrary fixed prime powers, have a joint uniform distribution. We give a proof with a power-saving error on every interval of consecutive nonnegative integers, uniform in the interval's starting point. A fixed congruence on the index may be imposed independently. The resulting statement for the part of a factorial coprime to an arbitrary fixed modulus gives the densities and relative density asked for in Problems 2 and 3 of Hajdu and Papp. The proof uses a finite digit-window identity, a strict local character contraction, and a carry estimate followed by finite Fourier analysis.

Provenance statement

The mathematical draft and exposition were produced primarily with OpenAI Codex under the direction of the listed contributors. Lean was used for supporting arithmetic and Fourier lemmas; the complete distribution theorem is not yet formally verified. No independent referee acceptance is claimed.

Tools used

OpenAI
Codex
Lean
Lean

References

  1. D. Berend and G. Kolesnik, Regularity of patterns in the factorization of n!n! , J. Number Theory 124 (2007), no. 1, 181--192. doi:10.1016/j.jnt.2006.08.010 (https://doi.org/10.1016/j.jnt.2006.08.010).DOI
  2. L. Hajdu and Á. Papp, On asymptotic density properties of the sequence (n!)n=0∞(n!)_{n=0}^{\infty} , Acta Arith. 184 (2018), no. 4, 317--340. doi:10.4064/aa170410-21-8 (https://doi.org/10.4064/aa170410-21-8).DOI

Version history

  1. v1 · 7 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 11, 2026