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TIGHT UNIVERSAL SUMS OF m-GONAL NUMBERS

Part of: Diophantine equations

Published online by Cambridge University Press:  13 July 2022

JANGWON JU Open the ORCID record for JANGWON JU [Opens in a new window]  and
MINGYU KIM Open the ORCID record for MINGYU KIM [Opens in a new window]
JANGWON JU
Affiliation:
Department of Mathematics, University of Ulsan, Ulsan 44610, Korea e-mail: jangwonju@ulsan.ac.kr
MINGYU KIM*
Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
*
e-mail: kmg2562@skku.edu
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Abstract

For a positive integer n, let $\mathcal T(n)$ denote the set of all integers greater than or equal to n. A sum of generalised m-gonal numbers g is called tight $\mathcal T(n)$ -universal if the set of all nonzero integers represented by g is equal to $\mathcal T(n)$ . We prove the existence of a minimal tight $\mathcal T(n)$ -universality criterion set for a sum of generalised m-gonal numbers for any pair $(m,n)$ . To achieve this, we introduce an algorithm giving all candidates for tight $\mathcal T(n)$ -universal sums of generalised m-gonal numbers. Furthermore, we provide some experimental results on the classification of tight $\mathcal T(n)$ -universal sums of generalised m-gonal numbers.

Keywords

sums of polygonal numberstight universalescalation algorithm

MSC classification

Primary: 11D09: Quadratic and bilinear equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 1 , February 2023 , pp. 40 - 52
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The research of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1F1A1064037). The research of the second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2021R1C1C2010133).

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