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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
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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.
Small Solutions of ϕ1x12 + . . . + ϕnxn2 = 0
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- Journal:
- Canadian Journal of Mathematics / Volume 52 / Issue 3 / 01 June 2000
- Published online by Cambridge University Press:
- 20 November 2018, pp. 613-632
- Print publication:
- 01 June 2000
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- Article
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- You have access
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Let
${{\phi }_{1}},...,{{\phi }_{n}}\,(n\,\ge \,2)$ be nonzero integers such that the equation
$$\sum\limits_{i=1}^{n}{{{\phi }_{i}}x_{i}^{2}\,=\,0}$$is solvable in integers
${{x}_{1}},...,{{x}_{n}}$ not all zero. It is shown that there exists a solution satisfying,
$$0\,<\,\sum\limits_{i}^{n}{\left| {{\phi }_{i}}\left| x_{i}^{2}\,\le \,2 \right|{{\phi }_{1}}...{{\phi }_{n}}\, \right|}$$and that the constant 2 is best possible.
