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Small Solutions of ϕ1x12 + . . . + ϕnxn2 = 0
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 2000