{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:I5U2XY6VTDGT33PJCZLK7CFUWK","short_pith_number":"pith:I5U2XY6V","schema_version":"1.0","canonical_sha256":"4769abe3d598cd3dede91656af88b4b2a1d8f93b26a25028323de794ef4a9880","source":{"kind":"arxiv","id":"1612.01468","version":1},"attestation_state":"computed","paper":{"title":"Consecutive primes and Beatty sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Victor Z. Guo, William D. Banks","submitted_at":"2016-12-05T18:45:06Z","abstract_excerpt":"Fix irrational numbers $\\alpha,\\hat\\alpha>1$ of finite type and real numbers $\\beta,\\hat\\beta\\ge 0$, and let $B$ and $\\hat B$ be the Beatty sequences $$ B:=(\\lfloor\\alpha m+\\beta\\rfloor)_{m\\ge 1}\\quad\\text{and}\\quad\\hat B:=(\\lfloor\\hat\\alpha m+\\hat\\beta\\rfloor)_{m\\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\\sharp)$ of consecutive primes for which $p\\in B$ and $p^\\sharp\\in\\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \\big|\\{p\\le x:p\\in B\\text{ and }p^\\sharp\\in\\hat B\\}\\big|=(\\alpha\\hat\\alpha)^{-1}\\pi(x)+O\\big(x(\\log "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.01468","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-05T18:45:06Z","cross_cats_sorted":[],"title_canon_sha256":"dfa6641d5494ca0d3c0a5dfa1e484a2f10c3cc774f163f27f4cc2e04ae42e0a7","abstract_canon_sha256":"d12341da06a4c3bf7a2c3d3e38a254b479e2271164e89ba087b65fa9884e38ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:52.601301Z","signature_b64":"LngJRlKEAhoKmFH4lVfXd74MaPaKFx+riKmSfIrJj7X4HAFfdnAq061vOWGDk8Dn1FiaU8MnYT1n5dHE+dTQCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4769abe3d598cd3dede91656af88b4b2a1d8f93b26a25028323de794ef4a9880","last_reissued_at":"2026-05-18T00:55:52.600847Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:52.600847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Consecutive primes and Beatty sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Victor Z. Guo, William D. Banks","submitted_at":"2016-12-05T18:45:06Z","abstract_excerpt":"Fix irrational numbers $\\alpha,\\hat\\alpha>1$ of finite type and real numbers $\\beta,\\hat\\beta\\ge 0$, and let $B$ and $\\hat B$ be the Beatty sequences $$ B:=(\\lfloor\\alpha m+\\beta\\rfloor)_{m\\ge 1}\\quad\\text{and}\\quad\\hat B:=(\\lfloor\\hat\\alpha m+\\hat\\beta\\rfloor)_{m\\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\\sharp)$ of consecutive primes for which $p\\in B$ and $p^\\sharp\\in\\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \\big|\\{p\\le x:p\\in B\\text{ and }p^\\sharp\\in\\hat B\\}\\big|=(\\alpha\\hat\\alpha)^{-1}\\pi(x)+O\\big(x(\\log "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01468","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.01468","created_at":"2026-05-18T00:55:52.600913+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.01468v1","created_at":"2026-05-18T00:55:52.600913+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.01468","created_at":"2026-05-18T00:55:52.600913+00:00"},{"alias_kind":"pith_short_12","alias_value":"I5U2XY6VTDGT","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"I5U2XY6VTDGT33PJ","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"I5U2XY6V","created_at":"2026-05-18T12:30:22.444734+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK","json":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK.json","graph_json":"https://pith.science/api/pith-number/I5U2XY6VTDGT33PJCZLK7CFUWK/graph.json","events_json":"https://pith.science/api/pith-number/I5U2XY6VTDGT33PJCZLK7CFUWK/events.json","paper":"https://pith.science/paper/I5U2XY6V"},"agent_actions":{"view_html":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK","download_json":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK.json","view_paper":"https://pith.science/paper/I5U2XY6V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.01468&json=true","fetch_graph":"https://pith.science/api/pith-number/I5U2XY6VTDGT33PJCZLK7CFUWK/graph.json","fetch_events":"https://pith.science/api/pith-number/I5U2XY6VTDGT33PJCZLK7CFUWK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK/action/storage_attestation","attest_author":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK/action/author_attestation","sign_citation":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK/action/citation_signature","submit_replication":"https://pith.science/pith/I5U2XY6VTDGT33PJCZLK7CFUWK/action/replication_record"}},"created_at":"2026-05-18T00:55:52.600913+00:00","updated_at":"2026-05-18T00:55:52.600913+00:00"}