{"paper":{"title":"The primes contain arbitrarily long polynomial progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Tamar Ziegler, Terence Tao","submitted_at":"2006-10-01T18:32:13Z","abstract_excerpt":"We establish the existence of infinitely many \\emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \\in \\Z[\\m]$ in one unknown $\\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\\eps > 0$, we show that there are infinitely many integers $x,m$ with $1 \\leq m \\leq x^\\eps$ such that $x+P_1(m), ..., x+P_k(m)$ are simultaneously prime. The arguments are based on those in Green and Tao, which treated the linear case $P_i = (i-1)\\m$ and $\\eps=1$; the main new features are a localization of the shift parameters (and the attendant Gowers norm obj"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610050","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}