{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:NLFUQYI2KXK7H553UQMHVN22FD","short_pith_number":"pith:NLFUQYI2","schema_version":"1.0","canonical_sha256":"6acb48611a55d5f3f7bba4187ab75a28fefe4f38215cdfc2b60c92d42673d864","source":{"kind":"arxiv","id":"1908.04938","version":1},"attestation_state":"computed","paper":{"title":"A Constructive Proof of Masser's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander J. Barrios","submitted_at":"2019-08-14T03:21:56Z","abstract_excerpt":"The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $\\epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+\\epsilon}<\\max\\!\\left\\{ \\left\\vert c_{4}^{3}\\right\\vert,c_{6}^{2}\\right\\} $ where $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$ and $N_{E}$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_{E}^{6}<\\max\\!\\left\\{ \\left\\vert c_{4}^{3}\\right\\vert,c_{6}^{2}\\right\\} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion."},"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":"1908.04938","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-14T03:21:56Z","cross_cats_sorted":[],"title_canon_sha256":"051d7796f68e7dffc90ae4f1f432418871f896bdfbbc7b88fa7b0fc5e0da6e76","abstract_canon_sha256":"b029fc231b3f36fec414ef485e2641fef72989e14dbd08cb613dde43652001d1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:47:37.544773Z","signature_b64":"O/42FNdsHNsvXL4GpqMq2bOyRfFIpO/wH1/l4t9hnXzwQUk8zvzLolmo04L6G41a2V3vkBfvD/SdsoiJXdrUBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6acb48611a55d5f3f7bba4187ab75a28fefe4f38215cdfc2b60c92d42673d864","last_reissued_at":"2026-07-05T04:47:37.544340Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:47:37.544340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Constructive Proof of Masser's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander J. Barrios","submitted_at":"2019-08-14T03:21:56Z","abstract_excerpt":"The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $\\epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+\\epsilon}<\\max\\!\\left\\{ \\left\\vert c_{4}^{3}\\right\\vert,c_{6}^{2}\\right\\} $ where $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$ and $N_{E}$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_{E}^{6}<\\max\\!\\left\\{ \\left\\vert c_{4}^{3}\\right\\vert,c_{6}^{2}\\right\\} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04938","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.04938/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.04938","created_at":"2026-07-05T04:47:37.544400+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.04938v1","created_at":"2026-07-05T04:47:37.544400+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04938","created_at":"2026-07-05T04:47:37.544400+00:00"},{"alias_kind":"pith_short_12","alias_value":"NLFUQYI2KXK7","created_at":"2026-07-05T04:47:37.544400+00:00"},{"alias_kind":"pith_short_16","alias_value":"NLFUQYI2KXK7H553","created_at":"2026-07-05T04:47:37.544400+00:00"},{"alias_kind":"pith_short_8","alias_value":"NLFUQYI2","created_at":"2026-07-05T04:47:37.544400+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/NLFUQYI2KXK7H553UQMHVN22FD","json":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD.json","graph_json":"https://pith.science/api/pith-number/NLFUQYI2KXK7H553UQMHVN22FD/graph.json","events_json":"https://pith.science/api/pith-number/NLFUQYI2KXK7H553UQMHVN22FD/events.json","paper":"https://pith.science/paper/NLFUQYI2"},"agent_actions":{"view_html":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD","download_json":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD.json","view_paper":"https://pith.science/paper/NLFUQYI2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.04938&json=true","fetch_graph":"https://pith.science/api/pith-number/NLFUQYI2KXK7H553UQMHVN22FD/graph.json","fetch_events":"https://pith.science/api/pith-number/NLFUQYI2KXK7H553UQMHVN22FD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD/action/storage_attestation","attest_author":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD/action/author_attestation","sign_citation":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD/action/citation_signature","submit_replication":"https://pith.science/pith/NLFUQYI2KXK7H553UQMHVN22FD/action/replication_record"}},"created_at":"2026-07-05T04:47:37.544400+00:00","updated_at":"2026-07-05T04:47:37.544400+00:00"}