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Given any set $\\mathcal D$ consisting of finite strictly increasing sequences $(d_1,d_2,\\dots, d_l)$ of positive integers such that, for each prime integer $p$, the set $\\{p\\mathbb Z, d_1+p\\mathbb Z,\\dots, d_l+p\\mathbb Z\\}$ does not contain all the cosets modulo $p$, we can stipulate to have, for each $(d_1,\\dots, d_l)\\in \\mathcal D$, a cofinal set of progressions $(f, f+d_1, \\dots, f+d_l)$ of prime elements in our principal ideal domain $R_\\tau$. Moreover, we can simul"},"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":"2204.06866","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-04-14T10:38:31Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"59a3241902d4b12104067a1f6fe0051d77ab39b16343db73b3731fea12e22d3a","abstract_canon_sha256":"93f07c3fee3bc45df157ec4dd9d42442006057e134a807ea4173515f115d7422"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:18:46.617498Z","signature_b64":"CYI9zl+ThpJB3DS2eemLbn0dgHmri54Iut+8ig9bdMgu0b/ezckDQ/T9aH6KMs8imtmIDJ0QYDwM41f/53txCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb27d37997cce12c6aefda3b1efde0cef636441ff4dba525afe98978b2ba2b02","last_reissued_at":"2026-07-05T05:18:46.616566Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:18:46.616566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\\mathbb Q[x]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Ester Sgallov\\'a, Jana Glivick\\'a, Jan \\v{S}aroch","submitted_at":"2022-04-14T10:38:31Z","abstract_excerpt":"We show how to construct discretely ordered principal ideal subrings of $\\mathbb Q[x]$ with various types of prime behaviour. Given any set $\\mathcal D$ consisting of finite strictly increasing sequences $(d_1,d_2,\\dots, d_l)$ of positive integers such that, for each prime integer $p$, the set $\\{p\\mathbb Z, d_1+p\\mathbb Z,\\dots, d_l+p\\mathbb Z\\}$ does not contain all the cosets modulo $p$, we can stipulate to have, for each $(d_1,\\dots, d_l)\\in \\mathcal D$, a cofinal set of progressions $(f, f+d_1, \\dots, f+d_l)$ of prime elements in our principal ideal domain $R_\\tau$. 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