Algorithms for representing positive odd integers as the sum of arithmetic progressions

Peter J.-S. Shiue, Anthony G. Shannon, Shen C. Huang, Michael R. Schwob, Rama Venkat
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 665–680
DOI: 10.7546/nntdm.2024.30.4.665-680
Full paper (PDF, 226 Kb)

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Authors and affiliations

Peter J.-S. Shiue
Department of Mathematical Sciences, University of Nevada, Las Vegas
4505 S Maryland Pkwy, Las Vegas, NV 89154, United States

Anthony G. Shannon
Warrane College, University of New South Wales
Kensington, NSW 2033, Australia

Shen C. Huang
Department of Mathematical Sciences, University of Nevada, Las Vegas
4505 S Maryland Pkwy, Las Vegas, NV 89154, United States

Michael R. Schwob
Department of Statistics and Data Sciences, University of Texas, Austin
110 Inner Campus Drive Austin, TX 78712, United States

Rama Venkat
College of Engineering, University of Nevada, Las Vegas
4505 S Maryland Pkwy, Las Vegas, NV 89154, United States

Abstract

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

Keywords

• Arithmetic progressions
• Balancing
• Fermat
• Fibonacci
• Geometric
• Jacobsthal
• Leonardo
• Lucas
• Mersenne
• Padovan
• Pell
• Perrin sequences

2020 Mathematics Subject Classification

• 11B39
• 11A25

References

1. Atanassov, K. T., Atanassova, V., Shannon, A. G., & Turner, J. C. (2002). New Visual Perspectives on Fibonacci Numbers. New York/Singapore: World Scientific.
2. Atanassov, K. T., Knott, R., Ozeki, K., Shannon, A. G., & Szalay, L. (2003). Inequalities among related pairs of Fibonacci numbers. The Fibonacci Quarterly, 41(1), 20–22.
3. 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.
4. Ho, C., He, T.-X., Shiue, & P. J.-S. (2023). Representations of positive integers as sums of arithmetic progressions, II. Notes on Number Theory and Discrete Mathematics, 29(2), 260–275.
5. Hoggatt Jr, V. E. (1969). Fibonacci and Lucas Numbers. Boston: Houghton Mifflin.
6. Junaidu, S. B., Laradji, A., & Umar, A. (2010). Powers of integers as sums of consecutive odd numbers. The Mathematical Gazette, 94(529), 117–119.
7. Long, C., Cohen, G. L., Langtry, T., & Shannon, A. G. (1993). Arithmetic sequences and second order recurrences. In: Bergum, G. E., Philippou, A. N., & Horadam, A. F. (Eds.). Applications of Fibonacci Numbers, 5, 449–457. Dordrecht: Kluwer.
8. Mahanthappa, M. (1991). Arithmetic sequences and Fibonacci quadratics. The Fibonacci Quarterly, 29, 343–346.
9. Munagi, A. O., & Shonhiwa, T. (2008). On the partitions of a number into arithmetic progressions. Journal of Integer Sequences, 11(5), Article ID 08.5.34.
10. Munagi, A. O., & de Vega, F. J. (2023). An extension of Sylvester’s theorem on arithmetic progressions. Symmetry, 15(6), Article ID 1276.
11. Nagell, T. (2021). Introduction to Number Theory, Vol. 163, American Mathematical Society.
12. Shiue, P. J., Huang, S. C., & Jameson, E. (2020). On algorithms for computing the sums of powers of arithmetic progressions. Notes on Number Theory and Discrete Mathematics, 26(4), 113–121.
13. Shiue, P. J., Huang, S. C., & Reyes, J. E. (2021). Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers. Notes on Number Theory and Discrete Mathematics, 27(4), 140–148.
14. Sylvester, J. J., & Franklin, F. (1882). A constructive theory of partitions, arranged in three acts, an interact and an exodion. American Journal of Mathematics, 5(1), 251–330.
15. Wheatstone, C. (1844). Beschreibung verschiedener neuen Instrumente und Methoden zur Bestimmung der Constanten einer Volta’schen Kette. Annalen der Physik, 138(8), 499–543.

Manuscript history

• Received: 22 May 2024
• Accepted: 3 October 2024
• Online First: 5 November 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).