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


Published:20000601
Printed: Jun 2000
Zhiming M. Ou
Kenneth S. Williams
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.