Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T16:31:26.387Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
DAEJUN KIM*
Affiliation:
Research Institute of Mathematics, Seoul National University, Seoul08826, Korea
KYOUNGMIN KIM
Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon16419, Korea e-mail: [email protected]
MINGYU KIM
Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon16419, Korea e-mail: [email protected]
BYEONG-KWEON OH
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul08826, Korea e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

Bhargava, M., ‘On the Conway–Schneeberger fifteen theorem’, Contemp. Math. 272 (2000), 27–38.10.1090/conm/272/04395CrossRefGoogle Scholar
Doyle, G. and Williams, K. S., ‘Prime-universal quadratic forms $a{x}^2+b{y}^2+c{z}^2$ and $a{x}^2+b{y}^2+c{z}^2+d{w}^2$ ’, Bull. Aust. Math. Soc. 101 (2020), 1–12.10.1017/S0004972719001023CrossRefGoogle Scholar
Ju, J., Oh, B.-K. and Seo, B., ‘Ternary universal sums of generalized polygonal numbers’, Int. J. Number Theory 15 (2019), 655–675.10.1142/S1793042119500350CrossRefGoogle Scholar
Kitaoka, Y., Arithmetic of Quadratic Forms , Cambridge Tracts in Mathematics, 106 (Cambridge University Press, Cambridge, 1993).10.1017/CBO9780511666155CrossRefGoogle Scholar
Oh, B.-K., ‘Regular positive ternary quadratic forms’, Acta Arith. 147 (2011), 233–243.10.4064/aa147-3-3CrossRefGoogle Scholar
Oh, B.-K., ‘Ternary universal sums of generalized pentagonal numbers’, J. Korean Math. Soc. 48 (2011), 837–847.10.4134/JKMS.2011.48.4.837CrossRefGoogle Scholar
O’Meara, O. T., Introduction to Quadratic Forms (Springer Verlag, New York, 1963).10.1007/978-3-642-62031-7CrossRefGoogle Scholar