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arXiv · 2610.03417

Sum-product patterns in the shifted primes

Florian K. Richter·Joni Teräväinen
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Abstract

We show that the set $\mathbb{P}-1$ of shifted primes contains infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ with $x,y$ arbitrarily large distinct integers. More strongly, we can also show that, for any $k\geq 1$, the set $\mathbb{P}-1$ contains longer patterns of the form $\{x,x+y,\ldots, x+ky,xy\}$ with $x,y$ arbitrarily large distinct integers, a statement that contains the Green--Tao theorem as a special case.

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Florian K. Richter, Joni Teräväinen. 2026-10-02. Sum-product patterns in the shifted primes. https://arxiv.org/abs/2610.03417

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