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SUMS OF SQUARES OF INTEGERS EXCEPT FOR A FIXED ONE

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  20 November 2025

WONJUN CHAE Open the ORCID record for WONJUN CHAE [Opens in a new window] ,
YUN-SEONG JI ,
KISUK KIM ,
KYOUNGMIN KIM Open the ORCID record for KYOUNGMIN KIM [Opens in a new window] ,
BYEONG-KWEON OH Open the ORCID record for BYEONG-KWEON OH [Opens in a new window]  and
JONGHEUN YOON Open the ORCID record for JONGHEUN YOON [Opens in a new window]
WONJUN CHAE
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University , Seoul 08826, Korea e-mail: wonjun97@snu.ac.kr
YUN-SEONG JI
Affiliation:
Research Institute of Mathematics, Seoul National University , Seoul 08826, Korea e-mail: ys0430@snu.ac.kr
KISUK KIM
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University , Seoul 08826, Korea e-mail: samsungkks@snu.ac.kr
KYOUNGMIN KIM
Affiliation:
Department of Mathematics, Hannam University , Daejeon 34430, Korea e-mail: kiny30@hnu.kr
BYEONG-KWEON OH
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University , Seoul 08826, Korea e-mail: bkoh@snu.ac.kr
JONGHEUN YOON*
Affiliation:
Faculty of Mathematics and Physics, Department of Algebra, Charles University , Sokolovská 83, 186 00 Praha 8, Czech Republic and Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
*
e-mail: jongheun.yoon@matfyz.cuni.cz
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Abstract

We study sums of squares of integers except for a fixed one. For any nonnegative integer n, we find the minimum number of squares of integers except for n whose sums represent all positive integers that are represented by a sum of squares except for n. This problem could be considered as a generalisation of the result of Dubouis [‘Solution of a problem of J. Tannery’, Intermédiaire Math. 18 (1911), 55–56] for the case when $n=0$.

Keywords

sums of squares

MSC classification

Primary: 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 11
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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Footnotes

The first and fifth authors were supported by National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (NRF-2020R1A5A1016126 and RS-2024-00342122); the second author was supported by National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (NRF-2021R1I1A1A01043959 and RS-2024-00342122); the sixth author was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00342122) and by the Czech Science Foundation (GAČR) (grant 21-00420M).

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