Representations of positive integers as sums of arithmetic progressions, II

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 260–275
DOI: 10.7546/nntdm.2023.29.2.260-275
Full paper (PDF, 265 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

As mentioned in the first part of this paper, our paper was motivated by two classical papers on the representations of integers as sums of arithmetic progressions. One of them is a paper by Sir Charles Wheatstone and the other is a paper by James Joseph Sylvester. Part I of the paper, though contained some extensions of Wheatstone’s work, was primarily devoted to extensions of Sylvester’s Theorem. In this part of the paper, we will pay more attention on the problems initiated by of Wheatstone on the representations of powers of integers as sums of arithmetic progressions and the relationships among the representations for different powers of the integer. However, a large part in this portion of the paper will be devoted to the extension of a clever method recently introduced by S. B. Junaidu, A. Laradji, and A. Umar and the problems related to the extension. This is because that this extension, not only will be our main tool for study ing the relationships of the representations of different powers of an integer, but also seems to be interesting in its own right. In the process of doing this, we need to use a few results from the first part of the paper. On the other hand, some of our results in this part will also provide certain new information on the problems studied in the first part. However, for readers who are interested primarily in the results of this part, we have repeated some basic facts from Part I of the paper so that the reader can read this part independently from the first part.

Keywords

• Representation by arithmetic progressions
• Junaidu–Laradji–Umar process
• Inducement of representations
• Complementary factors

2020 Mathematics Subject Classification

• 11B25
• 11A41

References

1. Apostol, T. M. (2003). Sum of consecutive positive integers. The Mathematical Gazette, 87(508), 98–101.
2. Carlitz, L. (1962). Generating functions for powers of a certain sequences of numbers. /Duke Mathematical Journal, 29, 521–537.
3. Dickson, L. E. (1923), (1966). History of the Theory of Numbers, Vol. II, Carnegie Institution of Washington, Washington, D.C., Vol. I 1919; Vol. II, 1920, Vol. III, 1923, reprinted by Chelsea, New York, 1966.
4. Ho, C., He, T.-X., & Shiue, P. J.-S. (2023). Representations of positive integers as sums of arithmetic progressions, I. Notes on Number Theory and Discrete Mathematics, 29(2), 241–259.
5. Horadam, A. F. (1965), Generating functions for powers of a certain generalised sequence of numbers. Duke Mathematical Journal, 32, 437–446.
6. Junaidu, S. B., Laradje, A., & Umar, A. (2010). Powers of integers as sums of consecutive odd integers. The Mathematical Gazette, 94, 117–119.
7. Sylvester, J. J. (1882). A constructive theory of partitions, arranged in three acts, an interact and an exodion. American Journal of Mathematics, 5(1), 251–330.
8. Wheatstone, C. (1854-1855). On the formation of powers from arithmetical progressions. Proceedings of the Royal Society of London, 7, 145–151.

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).