{"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"}