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Furthermore there are infinitely many positive integers $M$ that can be written in $n$ different ways, $n\\in\\{2,3\\}$, as the sum of three distinct positive integers with the same product $N$ and $E_{(M,N)}(\\Q)$ has rank at least $n$."},"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":"1303.6705","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-26T23:38:26Z","cross_cats_sorted":[],"title_canon_sha256":"52fa5266adf1f99f65915310193cf4cd52f6d7fcf315400cec29cb8105a72504","abstract_canon_sha256":"0bad54d55c34a997c87f36c46a4fcfb5cf6f9792fda1b8dcce7b189cbfa9a1ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:48.232985Z","signature_b64":"MN6Fv44hQQBGvvhAQJiWLYYI38PJ7sGaR166XIJ2PrxOOWMhotod4idJ/eHx9kbF6HY2KXlCoNevn9qL9JRxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed0d8a2b4537bf3df1a7645a7c39ee94c55f0bdaf9b16067b814413a091b27fe","last_reissued_at":"2026-05-18T02:16:48.232286Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:48.232286Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partitions with equal products and elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad Sadek, Nermine El-Sissi","submitted_at":"2013-03-26T23:38:26Z","abstract_excerpt":"Let $a,b,c$ be distinct positive integers. 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