PRIME-UNIVERSAL DIAGONAL QUADRATIC FORMS

Abstract

AbstractA (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.