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Using methods similar to theirs, we show that $p_{k-1}< (2N)^{1/5}$ and that $p_{k-1}p_k < 6^{1/4}N^{1/2}.$ We also show that if $p_k$ and $p_{k-1}$ are close to each other than these bounds can be further strengthened."},"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":"1810.11734","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-28T00:03:51Z","cross_cats_sorted":[],"title_canon_sha256":"2c78583c5365e215008ab28461fb1d699d0f761d15b888a6e20808eeda15fccb","abstract_canon_sha256":"185ef8fe064f4cfed8af9b49567fcac6bd5bd88d0522639b2277ef020f97f575"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:13.491725Z","signature_b64":"v1EOGGOwMajNbs4M1hN8fU7nf54JbOdtGiZXzJ7MZrWSIgGvrTA7sQ4THKdp7C13ujfu3Ijb5+FEOfTyu9KIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be5f0ad4f2d5014cf85415734385c08e55ef5ae5eddc5090b76b48007f1a86a9","last_reissued_at":"2026-05-17T23:58:13.491202Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:13.491202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper bounds on the second largest prime factor of an odd perfect number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Joshua Zelinsky","submitted_at":"2018-10-28T00:03:51Z","abstract_excerpt":"Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N= p_1^{a_1}p_2^{a_2} \\cdots p_k^{a_k}$ where $p_1 < p_2 \\cdots < p_k$ then one must have $p_k < 3^{1/3}N^{1/3}$. 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