No CrossRef data available.
Article contents
Small Solutions of ϕ1x12 + . . . + ϕnxn2 = 0
Published online by Cambridge University Press: 20 November 2018
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
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.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2000
References
[1]
Birch, B. J. and Davenport, H., Quadratic equations in several variables. Proc. Cambridge Philos. Soc. 54(1958), 135–138.Google Scholar
[2]
Borevich, Z. I. and Shafarevich, I. R., Number Theory. Academic Press, New York and London, 1966.Google Scholar
[3]
Keller, O.-H., Geometrie der Zahlen. Enzyklopädie der Mathematischen Wissenschaften, Band I2, Heft 11, Teil III, B. G. Teubner, Leipzig, 1954.Google Scholar
[4]
Mordell, L. J., On the magnitude of the integer solutions of the equation ax2 + by 2 + cz2 = 0. J. Number Theory
1(1969), 1–3.Google Scholar
[5]
Williams, K. S., On the size of a solution of Legendre's equation. Utilitas Math. 34(1988), 65–72.Google Scholar
You have
Access