On b-repdigit polygonal numbers

Adriana Mora and Eric Bravo
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 819–828
DOI: 10.7546/nntdm.2025.31.4.819-828
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Authors and affiliations

Adriana Mora
Departamento de Matemáticas, Universidad del Cauca
Calle 5 # 4-70, Colombia

Eric Bravo
Departamento de Matemáticas, Universidad del Cauca
Calle 5 # 4-70, Colombia

Abstract

We prove a finiteness theorem concerning repdigits in base b\ge 2 represented by a fixed quadratic polynomial. We also show that there is a finite number of polygonal numbers that are also b-repdigits for all b\ge 2 provided that (b,s)\notin \left\{\left(\frac{8(s-2)}{(s-4)^{2}}d+1,s\right):s\in [3,13]-\{4\}\right\}, where s\ge 3 denotes the number of sides of the polygon and d\in \{1,2,\ldots,b-1\}. We illustrate this result by finding all triangular, pentagonal and heptagonal numbers that are also b-repdigits for b\in [2,9]. This paper is motivated by a previous work of Kafle, Luca, and Togbé who considered the same finiteness problem for b=10 to find all pentagonal and heptagonal numbers that are also repdigits.

Keywords

  • Repdigit
  • Polygonal number
  • Bachet equation
  • Mordell curve

2020 Mathematics Subject Classification

  • 11Y50
  • 14H52

References

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Manuscript history

  • Received: 14 April 2025
  • Revised: 29 October 2025
  • Accepted: 5 November 2025
  • Online First: 8 November 2025

Copyright information

Ⓒ 2025 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).

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Cite this paper

Mora, A., & Bravo, E. (2025). On b-repdigit polygonal numbers. Notes on Number Theory and Discrete Mathematics, 31(4), 819-828, DOI: 10.7546/nntdm.2025.31.4.819-828.

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