A bound for repeated binomial coefficients having large prime power factors

Zachary King and R. Scott Williams
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 430–438
DOI: 10.7546/nntdm.2026.32.2.430-438
Full paper (PDF, 390 Kb)

Details

Authors and affiliations

Zachary King
Department of Mathematics & Statistics, University of Central Oklahoma
100 North University Drive – Box 129, Edmond, OK, 73034, United States

R. Scott Williams
Department of Mathematics & Statistics, University of Central Oklahoma
100 North University Drive – Box 129, Edmond, OK, 73034, United States

Abstract

If is an integer, we let denote the number of ways of expressing as a binomial coefficient. We illustrate a method of bounding by utilizing properties of Pascal’s Triangle under modular arithmetic. We use this method to show that is bounded for with proportionately “large” prime power factors, and we describe how it may be possible to derive stronger bounds.

Keywords

• Pascal’s Triangle
• Binomial coefficients
• Singmaster’s Conjecture
  Prime power
• Asymptotics
• Combinatorics

2020 Mathematics Subject Classification

• 11B75
• 11P32

References

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Manuscript history

• Received: 18 October 2025
• Revised: 13 July 2026
• Accepted: 27 August 2026
• Online First: 31 August 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).