Divisibility of the sums of the power of consecutive integers

Tian-Xiao He and Peter J.-S. Shiue
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 3, Pages 557–574
DOI: 10.7546/nntdm.2024.30.3.557-574
Full paper (PDF, 253 Kb)

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Authors and affiliations

Tian-Xiao He
Department of Mathematics, Illinois Wesleyan University
Bloomington, Illinois 61702, United States

Peter J.-S. Shiue
Department of Mathematical Sciences, University of Nevada, Las Vegas
Vegas, Nevada, 89154-4020, United States

Abstract

We study the divisibility of the sums of the odd power of consecutive integers, and for odd integers and , by using the Girard–Waring identity. Faulhaber’s approach for the divisibilities is discussed. Some expressions of power sums in terms of Stirling numbers of the second kind are represented.

Keywords

• Divisibility
• Sum of powers of consecutive integers
• Girard–Waring identity
• Recursive sequence
• Arithmetic series
• Faulhaber’s theorem

2020 Mathematics Subject Classification

• 05A15
• 11B99
• 11B83

References

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Manuscript history

• Received: 11 September 2024
• Accepted: 1 October 2024
• Online First: 3 October 2024

Copyright information

Ⓒ 2024 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).