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PRIME-UNIVERSAL QUADRATIC FORMS $ax^{2}+by^{2}+cz^{2}$ AND $ax^{2}+by^{2}+cz^{2}+dw^{2}$

Part of: Elementary number theory Forms and linear algebraic groups

Published online by Cambridge University Press:  27 September 2019

GREG DOYLE Open the ORCID record for GREG DOYLE [Opens in a new window]  and
KENNETH S. WILLIAMS Open the ORCID record for KENNETH S. WILLIAMS [Opens in a new window]
GREG DOYLE
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada email [email protected]
KENNETH S. WILLIAMS*
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada email [email protected]
*
✉[email protected]
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Abstract

A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$ there are integers $x_{1},\ldots ,x_{n}$ such that $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}=p$. We determine all possible prime-universal ternary quadratic forms $ax^{2}+by^{2}+cz^{2}$ and all possible prime-universal quaternary quadratic forms $ax^{2}+by^{2}+cz^{2}+dw^{2}$. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature.

Keywords

ternary quadratic formsquaternary quadratic formsprime-universality

MSC classification

Primary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
Secondary: 11A41: Primes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 1 , February 2020 , pp. 1 - 12
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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