Translation invariant quadratic forms and dense sets of primes

Lilu Zhao

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arXiv · 2105.12958

Translation invariant quadratic forms and dense sets of primes

Lilu Zhao

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

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Lilu Zhao. 2021-05-27. Translation invariant quadratic forms and dense sets of primes. https://arxiv.org/abs/2105.12958

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