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PRIME-UNIVERSAL DIAGONAL QUADRATIC FORMS

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  05 October 2020

JANGWON JU Open the ORCID record for JANGWON JU [Opens in a new window] ,
DAEJUN KIM Open the ORCID record for DAEJUN KIM [Opens in a new window] ,
KYOUNGMIN KIM Open the ORCID record for KYOUNGMIN KIM [Opens in a new window] ,
MINGYU KIM Open the ORCID record for MINGYU KIM [Opens in a new window]  and
BYEONG-KWEON OH Open the ORCID record for BYEONG-KWEON OH [Opens in a new window]
JANGWON JU
Affiliation:
Department of Mathematics, University of Ulsan, Ulsan, 44610, Republic of Korea e-mail: jangwonju@ulsan.ac.kr
DAEJUN KIM*
Affiliation:
Research Institute of Mathematics, Seoul National University, Seoul08826, Korea
KYOUNGMIN KIM
Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon16419, Korea e-mail: kiny30@skku.edu
MINGYU KIM
Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon16419, Korea e-mail: kmg2562@skku.edu
BYEONG-KWEON OH
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul08826, Korea e-mail: bkoh@snu.ac.kr
*
e-mail: goodkdj@snu.ac.kr
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Abstract

A (positive definite and integral) quadratic form is said to be prime-universal if it represents all primes. Recently, Doyle and Williams [‘Prime-universal quadratic forms $ax^2+by^2+cz^2$ and $ax^2+by^2+cz^2+dw^2$ ’, Bull. Aust. Math. Soc.101 (2020), 1–12] classified all prime-universal diagonal ternary quadratic forms and all prime-universal diagonal quaternary quadratic forms under two conjectures. We classify all prime-universal diagonal quadratic forms regardless of rank, and prove the so-called 67-theorem for a diagonal quadratic form to be prime-universal.

Keywords

diagonal quadratic formsprime-universal

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
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 3 , June 2021 , pp. 390 - 404
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (NRF-2019R1F1A1064037), the third author was supported by an NRF grant funded by MSIT (NRF-2020R1I1A1A01055225), the fourth author was supported by the NRF (NRF-2019R1A6A3A01096245) and the fifth author was supported by the NRF grant funded by MSIT (NRF-2020R1A5A1016126).

References

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