{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:F4TMXTXJJG7MS3YTNNX4TEZOOU","short_pith_number":"pith:F4TMXTXJ","canonical_record":{"source":{"id":"1908.09483","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","cross_cats_sorted":[],"title_canon_sha256":"709f66c441bcaccb1c94f96434b1f642748638f7236fd6011044cb49ad651b1c","abstract_canon_sha256":"e6dc03950a039c17f3d28730bc9a25da75970fe417355e03067ea2834675f266"},"schema_version":"1.0"},"canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","source":{"kind":"arxiv","id":"1908.09483","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09483","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09483v3","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09483","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"pith_short_12","alias_value":"F4TMXTXJJG7M","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"pith_short_16","alias_value":"F4TMXTXJJG7MS3YT","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"pith_short_8","alias_value":"F4TMXTXJ","created_at":"2026-07-05T01:08:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:F4TMXTXJJG7MS3YTNNX4TEZOOU","target":"record","payload":{"canonical_record":{"source":{"id":"1908.09483","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","cross_cats_sorted":[],"title_canon_sha256":"709f66c441bcaccb1c94f96434b1f642748638f7236fd6011044cb49ad651b1c","abstract_canon_sha256":"e6dc03950a039c17f3d28730bc9a25da75970fe417355e03067ea2834675f266"},"schema_version":"1.0"},"canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:08:24.528595Z","signature_b64":"XEVlIQhviV7Ah9R5XVdP9QKxXL0YYkplFL4bkT0ygF4Egy2zA6MyKI9ufPMW+zqNX786d8VDZ8JJDf5BVVM4Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","last_reissued_at":"2026-07-05T01:08:24.528139Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:08:24.528139Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.09483","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:08:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"43fSEqSft91VG2Xufw5YLCmU9v+8KODuz/NOeIDkL9RN9ju4R0t5GKt9LXTuWVI6qsvyPavp/UJlID7GdUOnDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T17:44:52.991068Z"},"content_sha256":"8cda1174eef253147b52b005159bedb5e3d579ccd5b7cb2d96823904ac8d3432","schema_version":"1.0","event_id":"sha256:8cda1174eef253147b52b005159bedb5e3d579ccd5b7cb2d96823904ac8d3432"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:F4TMXTXJJG7MS3YTNNX4TEZOOU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Numerical Semigroups generated by Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anton Rechenauer, Michael Hellus, Rolf Waldi","submitted_at":"2019-08-26T05:54:26Z","abstract_excerpt":"Let $p_1=2, p_2=3, p_3=5, \\ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that\n  $\\bullet$ $u_n\\sim3p_n$.\n  Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that\n  $\\bullet$ $f_n$ is an odd number for $n\\geq5$ and\n  $\\bullet$ $f_n\\sim3p_n$; further\n  $\\bullet$ $4p_n>f_{n+1}$ for $n\\geq1$.\n  If $f_n$ is odd for large $n$, then $f_n\\sim3p_n$. In case $f_n\\sim3p_n$ every large even integer $x$ is the sum of two"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09483","kind":"arxiv","version":3},"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/1908.09483/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:08:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mLzFmtjnznnuj/mp/QSIkfR7m+RWvdWpLzeLZRAai69/bJgOwqfcbmYUAZED9cNdtbdUdvTCIdCMMgcn73gHAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T17:44:52.991575Z"},"content_sha256":"ac8d0041b39ad3b092a02317066001984325f3b30e718a5ca414215108d4de27","schema_version":"1.0","event_id":"sha256:ac8d0041b39ad3b092a02317066001984325f3b30e718a5ca414215108d4de27"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/bundle.json","state_url":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-16T17:44:52Z","links":{"resolver":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU","bundle":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/bundle.json","state":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:F4TMXTXJJG7MS3YTNNX4TEZOOU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e6dc03950a039c17f3d28730bc9a25da75970fe417355e03067ea2834675f266","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","title_canon_sha256":"709f66c441bcaccb1c94f96434b1f642748638f7236fd6011044cb49ad651b1c"},"schema_version":"1.0","source":{"id":"1908.09483","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09483","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09483v3","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09483","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"pith_short_12","alias_value":"F4TMXTXJJG7M","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"pith_short_16","alias_value":"F4TMXTXJJG7MS3YT","created_at":"2026-07-05T01:08:24Z"},{"alias_kind":"pith_short_8","alias_value":"F4TMXTXJ","created_at":"2026-07-05T01:08:24Z"}],"graph_snapshots":[{"event_id":"sha256:ac8d0041b39ad3b092a02317066001984325f3b30e718a5ca414215108d4de27","target":"graph","created_at":"2026-07-05T01:08:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.09483/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $p_1=2, p_2=3, p_3=5, \\ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that\n  $\\bullet$ $u_n\\sim3p_n$.\n  Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that\n  $\\bullet$ $f_n$ is an odd number for $n\\geq5$ and\n  $\\bullet$ $f_n\\sim3p_n$; further\n  $\\bullet$ $4p_n>f_{n+1}$ for $n\\geq1$.\n  If $f_n$ is odd for large $n$, then $f_n\\sim3p_n$. In case $f_n\\sim3p_n$ every large even integer $x$ is the sum of two","authors_text":"Anton Rechenauer, Michael Hellus, Rolf Waldi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","title":"Numerical Semigroups generated by Primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09483","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8cda1174eef253147b52b005159bedb5e3d579ccd5b7cb2d96823904ac8d3432","target":"record","created_at":"2026-07-05T01:08:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e6dc03950a039c17f3d28730bc9a25da75970fe417355e03067ea2834675f266","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","title_canon_sha256":"709f66c441bcaccb1c94f96434b1f642748638f7236fd6011044cb49ad651b1c"},"schema_version":"1.0","source":{"id":"1908.09483","kind":"arxiv","version":3}},"canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","first_computed_at":"2026-07-05T01:08:24.528139Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:08:24.528139Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XEVlIQhviV7Ah9R5XVdP9QKxXL0YYkplFL4bkT0ygF4Egy2zA6MyKI9ufPMW+zqNX786d8VDZ8JJDf5BVVM4Cg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:08:24.528595Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.09483","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8cda1174eef253147b52b005159bedb5e3d579ccd5b7cb2d96823904ac8d3432","sha256:ac8d0041b39ad3b092a02317066001984325f3b30e718a5ca414215108d4de27"],"state_sha256":"2c305582700fb7cb2f3706f200e7162c540d84bcf5fafc4cfaf693af6d434e43"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y1U/Tf4BoPfOm6wgqlbOusufyaUKIlmVWBNhcW8U56LRib+jlbeJpxXar4NFmSh9ZXd3/qxSlIEj1y7W9SbLBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T17:44:52.997160Z","bundle_sha256":"51623e56e69edb34d12c4b87db178f36591653d66f31e1335e4d62fcbf1446b8"}}