Integral points on quadrics with prime coordinates (Q652548)

Let \(f (x _{1}, \dots, x _{s })\) be a regular indefinite integral quadratic form, and \(t\) an integer. Denote by \(V\) the affine quadric \(f(\mathbf x) = t\), and by \(V(\mathbb P)\) the set of \(\mathbf x\in V\) whose coordinates are simultaneously prime. Suppose \(s \geq 10\) is even, under a technical condition, the author proves that \(V(\mathbb P)\) is Zariski dense in \(V\) if and only if there is no bad prime for \(V\).