Translation invariant quadratic forms in $8$ variables and dense sets of primes

Abstract

Let $Q$ be a translation invariant indefinite quadratic form in $8$ variables. Let $\mathcal{P}$ be the set of all primes. Let $\mathcal{A}\subset \mathcal{P}\cap \{1,\dots ,X\}$. By developing the recent work of Green, we prove that subject to a mild condition on the form $Q$ there exist pairwise distinct primes $p_1,\dots ,p_8\in \mathcal{A}$ satisfying $Q(p_1,\dots ,p_8)=0$ provided that \[\frac{|\mathcal{A}|}{X/\log X}\gg_{Q}(\log\log X)^{-c}\] for some absolute constant $c>0$.

arXiv comment24 pages