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Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$

  Published:2000-06-01
 Printed: Jun 2000
  • Zhiming M. Ou
  • Kenneth S. Williams
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Abstract

Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that the equation $$ \sum_{i=1}^n \phi_i x_i^2 = 0 $$ is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown that there exists a solution satisfying $$ 0 < \sum_{i=1}^n |\phi_i| x_i^2 \leq 2 |\phi_1 \cdots \phi_n|, $$ and that the constant 2 is best possible.
Keywords: small solutions, diagonal quadratic forms small solutions, diagonal quadratic forms
MSC Classifications: 11E25 show english descriptions Sums of squares and representations by other particular quadratic forms 11E25 - Sums of squares and representations by other particular quadratic forms
 

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