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We show that there exist infinitely many natural numbers $n,m$ such that $n+P_1(m),\\dots,n+P_k(m)$ are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case $P_1(0)=\\dots=P_k(0)=0$ (though it allowed for the top degree coefficients to coincide)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1603.07817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-25T04:09:25Z","cross_cats_sorted":[],"title_canon_sha256":"afffcefd96ffab12c394f2cc7e39c4629ff084717e8b5c13fe5c8e2042259eeb","abstract_canon_sha256":"0348744855646d01d396adfdfc5c34a574521e5b0454f2288a2f1a3ec3054c18"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:17.756759Z","signature_b64":"lHKO1oziAAyzdiK7jGzzCMZks9lUcmQEsVm6mU5cx6UOebQ9ON9FQdRy1+WeuouAMv9k5EuUYjbQ9rN7e3DBBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"007e22431e027676eb8729c98701e912f6306039157c3c372a5a6bfb217e53c1","last_reissued_at":"2026-05-18T01:18:17.756177Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:17.756177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polynomial patterns in the primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tamar Ziegler, Terence Tao","submitted_at":"2016-03-25T04:09:25Z","abstract_excerpt":"Let $P_1,\\dots,P_k \\colon {\\bf Z} \\to {\\bf Z}$ be polynomials of degree at most $d$ for some $d \\geq 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such that $n+P_1(m),\\dots,n+P_k(m)$ are all not divisible by $p$. 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