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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
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 27 September 2019, pp. 1-12
- Print publication:
- February 2020
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- Article
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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.
