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Sum-product patterns in the shifted primes
Abstract
We show that the set 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≥ 1, the set ℙ−1 contains longer patterns of the form {x,x+y,…, x+ky,xy} with x,y arbitrarily large distinct integers, a statement that contains the Green--Tao theorem as a special case.
Related
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BibTeX
@misc{richter2026sumproduct,
title = {{Sum-product patterns in the shifted primes}},
author = {Florian K. Richter and Joni Teräväinen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03417},
eprint = {2610.03417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03417},
}
RIS
TY - UNPB
TI - Sum-product patterns in the shifted primes
AU - Florian K. Richter
AU - Joni Teräväinen
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03417
UR - https://arxiv.org/abs/2610.03417
AB - We show that the set 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≥ 1, the set ℙ−1 contains longer patterns of the form {x,x+y,…, x+ky,xy} with x,y arbitrarily large distinct integers, a statement that contains the Green--Tao theorem as a special case.
ER -