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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 π« be the set of all primes. Let πβπ«β©{1,β¦,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 p1,β¦,p8β π satisfying Q(p1,β¦,p8)=0 provided that |π|/(X/log X)β«Q(loglog X)βc for some absolute constant c>0.
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BibTeX
@misc{pei2026translation,
title = {{Translation invariant quadratic forms in $8$ variables and dense sets of primes}},
author = {Zhaochen Pei and Maoru Tan and Weiwen Zhang and Lilu Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01091},
eprint = {2610.01091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01091},
}
RIS
TY - UNPB
TI - Translation invariant quadratic forms in $8$ variables and dense sets of primes
AU - Zhaochen Pei
AU - Maoru Tan
AU - Weiwen Zhang
AU - Lilu Zhao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01091
UR - https://arxiv.org/abs/2610.01091
AB - Let Q be a translation invariant indefinite quadratic form in $8$ variables. Let π« be the set of all primes. Let πβπ«β©{1,β¦,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,β¦,p_8β π satisfying Q(p_1,β¦,p_8)=0 provided that |π|/(X/log X)β«_Q(loglog X)^βc for some absolute constant c>0.
ER -