Translation invariant quadratic forms and dense sets of primes

TL;DR

This paper proves that dense subsets of primes contain solutions to certain translation invariant quadratic equations, extending understanding of prime patterns under specific algebraic conditions.

Contribution

It establishes the existence of solutions to translation invariant quadratic forms within dense prime subsets, under a rank condition, for sufficiently large ranges.

Findings

Solutions exist in dense prime subsets for large X

Density threshold is approximately X / log X

Applicable to quadratic forms with at least 10 variables

Abstract

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 $|\mathcal{A}|\ge \frac{X}{\log X} (\log\log X)^{-\frac{1}{80}}.$