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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: [email protected]
MINGYU KIM*
Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
*
e-mail: [email protected]
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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).

References

Bhargava, M., ‘On the Conway–Schneeberger fifteen theorem’, Contemp. Math. 272 (2000), 27–38.10.1090/conm/272/04395CrossRefGoogle Scholar
Bosma, W. and Kane, B., ‘The triangular theorem of eight and representation by quadratic polynomials’, Proc. Amer. Math. Soc. 141(5) (2013), 1473–1486.CrossRefGoogle Scholar
Chan, W. K. and Oh, B.-K., ‘Representations of integral quadratic polynomials’, in: Diophantine Methods, Lattices and Arithmetic Theory of Quadratic Forms, Contemporary Mathematics, 587 (eds. Chan, W. K. and Schulze-Pillot, R.) (American Mathematical Society, Providence, RI, 2013), 31–46.CrossRefGoogle Scholar
Conway, J. H., ‘Universal quadratic forms and the fifteen theorem’, Contemp. Math. 272 (2000), 23–26.CrossRefGoogle Scholar
Hsia, J. S., Kitaoka, Y. and Kneser, M., ‘Representations of positive definite quadratic forms’, J. reine angew. Math. 301 (1978), 132–141.Google Scholar
Ju, J., ‘Universal sums of generalized pentagonal numbers’, Ramanujan J. 51(3) (2020), 479–494.CrossRefGoogle Scholar
Ju, J. and Kim, M., ‘Tight universal octagonal forms’, Preprint, 2022, arXiv:2202.09304.Google Scholar
Ju, J. and Oh, B.-K., ‘Universal sums of generalized octagonal numbers’, J. Number Theory 190 (2018), 292–302.10.1016/j.jnt.2017.12.014CrossRefGoogle Scholar
Kane, B. and Liu, J., ‘Universal sums of $m$ -gonal numbers’, Int. Math. Res. Not. IMRN 2020(20) (2020), 6999–7036.10.1093/imrn/rnz003CrossRefGoogle Scholar
Kim, B. M., Kim, M.-H. and Oh, B.-K., ‘A finiteness theorem for representability of quadratic forms by forms’, J. reine angew. Math. 581 (2005), 23–30.CrossRefGoogle Scholar
Kim, K., Lee, J. and Oh, B.-K., ‘Minimal universality criterion sets on the representations of binary quadratic forms’, J. Number Theory 238 (2022), 37–59.CrossRefGoogle Scholar
Kim, M., ‘Tight universal triangular forms’, Bull. Aust. Math. Soc. 105(3) (2022), 372–384.CrossRefGoogle Scholar
Kim, M. and Oh, B.-K., ‘Tight universal quadratic forms’, Preprint, 2021, arXiv:2104.02440.Google Scholar
O’Meara, O. T., ‘The integral representations of quadratic forms over local fields’, Amer. J. Math. 80 (1958), 843–878.CrossRefGoogle Scholar
Ramanujan, S., ‘On the expression of a number in the form $a{x}^2+b{y}^2+c{z}^2+d{u}^2$ ’, Proc. Cambridge Phil. Soc. 19 (1917), 11–21.Google Scholar