Equations of two sets of consecutive square sums

P. J. Bush and K. V. Murphy
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 677–691
DOI: 10.7546/nntdm.2022.28.4.677-691
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Authors and affiliations

P. J. Bush
Department of Mathematics, Valley City State University
101 College St. SE, Valley City, ND, 58072, United States

K. V. Murphy
Department of Mathematics, Rogue Community College
101 S. Bartlett St., Medford, OR, 97501, United States

Abstract

In this paper we investigate equations featuring sums of consecutive square integers, such as 3^2 + 4^2 = 5^2, and 108^2 + 109^2 + 110^2 = 133^2 + 134^2. In general, for a sum of m+1 consecutive square integers, x^2 + (x+1)^2 + \cdots + (x+m)^2, there is a distinct set of m consecutive squares, (x+n)^2 + (x+(n+1))^2 + \cdots + (x+(n+(m-1)))^2, to which these are equal. We present a bootstrap method for constructing these equations, which yields solutions comprising an infinite two-dimensional array. We apply a similar method to constructing consecutive square sum equations involving m+2 terms on the left, and m terms on the right, formed from two distinct sets of consecutive squares separated one term to the left of the equals sign, such as 2^2 + 3^2 + 6^2 = 7^2.

Keywords

  • Consecutive squares
  • Sums of squares
  • Perfect squares
  • Finite sums

2020 Mathematics Subject Classification

  • 11D09
  • 11C08

References

  1. Alfred, U. (1964). Consecutive integers whose sum of squares is a perfect square.
    Mathematic Magazine, 37(1), 19–32.
  2. Beeckmans, L. (1994). Squares expressible as a sum of consecutive squares. The American Mathematical Monthly, 101, 437–442.
  3. Bennett, M. A., Patel, V., & Siksek, S. (2016). Perfect powers that are sums of consecutive cubes. Mathematika, 63, 230–249.
  4. Bremner, A., & Stroeker, R. J., & Tzanakis, N. (1997). On sums of consecutive squares. Journal of Number Theory, 62, 39–70.
  5. Freitag, H. T., & Phillips, G. M. (1996). On the sum of consecutive squares. Applications of Fibonacci Numbers, 6, 137–142.
  6. Patel, V., & Siksek, S. (2017). On powers that are sums of consecutive like powers. Research in Number Theory, 3, Article 2.

Manuscript history

  • Received: 3 April 2021
  • Revised: 22 August 2022
  • Accepted: 25 October 2022
  • Online First: 27 October 2022

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Cite this paper

Bush, P. J., & Murphy, K. V. (2022). Equations of two sets of consecutive square sums. Notes on Number Theory and Discrete Mathematics, 28(4), 677-691, DOI: 10.7546/nntdm.2022.28.4.677-691.

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