Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$ 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 formsCategory:11E25