Prime-universal diagonal quadratic forms

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 $\textit{prime-universal}$ 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, $67$-Theorem for a diagonal quadratic form to be prime-universal.