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TIGHT UNIVERSAL TRIANGULAR FORMS

Part of: Forms and linear algebraic groups Additive number theory; partitions

Published online by Cambridge University Press:  18 November 2021

MINGYU KIM Open the ORCID record for MINGYU KIM [Opens in a new window]
MINGYU KIM*
Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
*
e-mail: kmg2562@skku.edu
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Abstract

For a subset S of nonnegative integers and a vector $\mathbf {a}=(a_1,\ldots ,a_k)$ of positive integers, define the set $V^{\prime }_S(\mathbf {a})=\{ a_1s_1+\cdots +a_ks_k : s_i\in S\}-\{0\}$ . For a positive integer n, let $\mathcal T(n)$ be the set of integers greater than or equal to n. We consider the problem of finding all vectors $\mathbf {a}$ satisfying $V^{\prime }_S(\mathbf {a})=\mathcal T(n)$ when S is the set of (generalised) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers.

Keywords

tight universal formpolygonal number

MSC classification

Primary: 11E12: Quadratic forms over global rings and fields
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11P99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 372 - 384
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by a National Research Foundation of Korea (NRF) grant funded by the government of Korea (MSIT) (NRF-2021R1C1C2010133).

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