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PRIME-UNIVERSAL DIAGONAL QUADRATIC FORMS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 103 / Issue 3 / June 2021
- Published online by Cambridge University Press:
- 05 October 2020, pp. 390-404
- Print publication:
- June 2021
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- Article
- Export citation
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A (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.
