Prime-universal diagonal quadratic forms
Jangwon Ju, Daejun Kim, Kyoungmin Kim, Mingyu Kim, and Byeong-Kweon Oh

TL;DR
This paper classifies all prime-universal diagonal quadratic forms of any rank and proves a 67-Theorem characterizing such forms, extending previous classifications limited to ternary and quaternary cases.
Contribution
It provides a complete classification of prime-universal diagonal quadratic forms across all ranks and establishes a new criterion (67-Theorem) for their universality.
Findings
Classified all prime-universal diagonal quadratic forms of any rank.
Proved the 67-Theorem for prime-universality of diagonal quadratic forms.
Extended previous classifications beyond ternary and quaternary forms.
Abstract
A (positive definite and integral) quadratic form is said to be if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all prime-universal diagonal quaternary quadratic forms under two conjectures proposed by themselves. In this article, we classify all prime-universal diagonal quadratic forms regardless of ranks. Furthermore, we prove, so called, -Theorem for a diagonal quadratic form to be prime-universal.
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Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Advanced Mathematical Identities
