  Let $f(x_1,\ldots,x_s)$ be a translation invariant indefinite quadratic form
of integer coefficients with $s\ge 10$. Let $\mathcal{A}\subseteq
\mathcal{P}\cap \{1,2,\ldots,X\}$. Let $X$ be sufficiently large. Subject to a
rank condition, we prove that there exist distinct primes $p_1,\ldots,p_s\in
\mathcal{A}$ such that $f(p_1,\ldots,p_s)=0$ as soon as $|\mathcal{A}|\ge
\frac{X}{\log X} (\log\log X)^{-\frac{1}{80}}.$
