# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2021; 58(4): 849-871

Published online July 1, 2021 https://doi.org/10.4134/JKMS.j200290

## Even $2$-universal quadratic forms of rank $5$

Yun-Seong Ji, Myeong Jae Kim, Byeong-Kweon Oh

Seoul National University; Seoul National University; Seoul National University

### Abstract

A (positive definite integral) quadratic form is called {\it even $2$-universal} if it represents all even quadratic forms of rank $2$. In this article, we prove that there are at most $55$ even $2$-universal even quadratic forms of rank $5$. The proofs of even $2$-universalities of some candidates will be given so that exactly $20$ candidates remain unproven.

Keywords: Even $2$-universal quadratic form

MSC numbers: Primary 11E12, 11E20

Supported by: This work of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (NRF-2017R1A5A1015626) and the Ministry of Education (NRF-2019R1I1A1A01060756). This work of the second and the third author was supported by the National Research Foundation of Korea (NRF-2019R1A2C1086347 and NRF-2020R1A5A1016126).