Representations of positive integers as sums of arithmetic progressions, I

Chungwu Ho, Tian-Xiao He and Peter J.-S. Shiue
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 241–259
DOI: 10.7546/nntdm.2023.29.2.241-259
Full paper (PDF, 284 Kb)

Details

Authors and affiliations

Chungwu Ho
Department of Mathematics, Southern Illinois University at Edwardsville
and Evergreen Valley College
Current Address: 1545 Laurelwood Crossing Ter., San Jose, California 95138, United States

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

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

Abstract

This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.

Keywords

• Arithmetic progressions
• Sylvester theorem
• Complementary factors

2020 Mathematics Subject Classification

• 11B25
• 11A41

References

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Manuscript history

• Received: 21 October 2022
• Revised: 10 March 2023
• Accepted: 26 April 2023
• Online First: 27 April 2023

Copyright information

Ⓒ 2023 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).