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More generally, we prove that if $\\nu$ is any positive integer, then\n  $$ \\liminf_{n\\to \\infty} (q_{n+\\nu}-q_n) \\le C(\\nu) = \\nu e^{\\nu-\\gamma} (1+o(1)).$$\n  We also prove several other results on the representation of numbers with exactly two prime factors by linear forms."},"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":"math/0609615","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2006-09-21T19:05:55Z","cross_cats_sorted":[],"title_canon_sha256":"00dba92015fdd4a618aaeec083e37ff1d827869b3974922363cd74944d9ef52f","abstract_canon_sha256":"a5c93f1513ac0447954ca1a82c6a93ebef100976787c5496979ba2ca2cf773e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:45.503437Z","signature_b64":"bDBjKZGQS/6MzMxV9ZCMev6RXaPKNmtgwNWUNW5I6oj3ik/DMJLr6ebYp8Bv6NbdFR3QRA9s6snBRPKMF4foCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9be43a611d43105feb2e2fb79c93d8fe7ec3f0cada3a55617a27464ad21e5406","last_reissued_at":"2026-05-18T02:57:45.502989Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:45.502989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Small gaps between products of two primes","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"C.Y. Yildirim, D. A. Goldston, J. Pintz, S.W. Graham","submitted_at":"2006-09-21T19:05:55Z","abstract_excerpt":"Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that\n  $$\\liminf_{n\\to \\infty} (q_{n+1}-q_n) \\le 6.$$\n  This sharpens an earlier result of the authors (arXivMath NT/0506067), which had 26 in place of 6. More generally, we prove that if $\\nu$ is any positive integer, then\n  $$ \\liminf_{n\\to \\infty} (q_{n+\\nu}-q_n) \\le C(\\nu) = \\nu e^{\\nu-\\gamma} (1+o(1)).$$\n  We also prove several other results on the representation of numbers with exactly two prime factors by linear forms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609615","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":"math/0609615","created_at":"2026-05-18T02:57:45.503066+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0609615v1","created_at":"2026-05-18T02:57:45.503066+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609615","created_at":"2026-05-18T02:57:45.503066+00:00"},{"alias_kind":"pith_short_12","alias_value":"TPSDUYI5IMIF","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"TPSDUYI5IMIF72ZO","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"TPSDUYI5","created_at":"2026-05-18T12:25:54.717736+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/TPSDUYI5IMIF72ZOF63ZZE6Y7Z","json":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z.json","graph_json":"https://pith.science/api/pith-number/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/graph.json","events_json":"https://pith.science/api/pith-number/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/events.json","paper":"https://pith.science/paper/TPSDUYI5"},"agent_actions":{"view_html":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z","download_json":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z.json","view_paper":"https://pith.science/paper/TPSDUYI5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0609615&json=true","fetch_graph":"https://pith.science/api/pith-number/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/graph.json","fetch_events":"https://pith.science/api/pith-number/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/action/storage_attestation","attest_author":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/action/author_attestation","sign_citation":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/action/citation_signature","submit_replication":"https://pith.science/pith/TPSDUYI5IMIF72ZOF63ZZE6Y7Z/action/replication_record"}},"created_at":"2026-05-18T02:57:45.503066+00:00","updated_at":"2026-05-18T02:57:45.503066+00:00"}