{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:GIW6H66WI75UPRZ5N2IVOGIY47","short_pith_number":"pith:GIW6H66W","canonical_record":{"source":{"id":"1401.0854","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-05T00:21:36Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"759253de15e3fa1220aae95dab31102bc8bc1b4dea77cee2bb93da12da88b9a6","abstract_canon_sha256":"8e30eff0999ff94693ad0c32461b7f5f09c9c51ed9db471886acec56b95f293d"},"schema_version":"1.0"},"canonical_sha256":"322de3fbd647fb47c73d6e91571918e7e7dd648edaaa0dbce19d3f8163911f12","source":{"kind":"arxiv","id":"1401.0854","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.0854","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"arxiv_version","alias_value":"1401.0854v2","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.0854","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"pith_short_12","alias_value":"GIW6H66WI75U","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GIW6H66WI75UPRZ5","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GIW6H66W","created_at":"2026-05-18T12:28:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:GIW6H66WI75UPRZ5N2IVOGIY47","target":"record","payload":{"canonical_record":{"source":{"id":"1401.0854","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-05T00:21:36Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"759253de15e3fa1220aae95dab31102bc8bc1b4dea77cee2bb93da12da88b9a6","abstract_canon_sha256":"8e30eff0999ff94693ad0c32461b7f5f09c9c51ed9db471886acec56b95f293d"},"schema_version":"1.0"},"canonical_sha256":"322de3fbd647fb47c73d6e91571918e7e7dd648edaaa0dbce19d3f8163911f12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:24.530553Z","signature_b64":"GA2E2h+MWElvHxqdfFrqxZRU2QdTqYL6jAA/KUsyKrBSOmSK1U36bkyQI+CkRCACYWZGdwu4XjTkOlhV0XIzBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"322de3fbd647fb47c73d6e91571918e7e7dd648edaaa0dbce19d3f8163911f12","last_reissued_at":"2026-05-18T01:22:24.529876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:24.529876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.0854","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:22:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yTZj+kbqM2JBHjAgXVw2XLqhQvkQ5dw+IpJsN3vQMb5XSPnqgjTQZ6+D7m7cK+3KYIIj9zT6eKXtf1t7MxSZBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:15:52.473765Z"},"content_sha256":"6260b971d001e132f220f6fc3770d565a89342621161efce9a96f355ef69975c","schema_version":"1.0","event_id":"sha256:6260b971d001e132f220f6fc3770d565a89342621161efce9a96f355ef69975c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:GIW6H66WI75UPRZ5N2IVOGIY47","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multivariate Ap\\'ery numbers and supercongruences of rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Armin Straub","submitted_at":"2014-01-05T00:21:36Z","abstract_excerpt":"One of the many remarkable properties of the Ap\\'ery numbers $A (n)$, introduced in Ap\\'ery's proof of the irrationality of $\\zeta (3)$, is that they satisfy the two-term supercongruences \\begin{equation*}\n  A (p^r m) \\equiv A (p^{r - 1} m) \\pmod{p^{3 r}} \\end{equation*} for primes $p \\geq 5$. Similar congruences are conjectured to hold for all Ap\\'ery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Ap\\'ery numbers by showing that they extend to all Taylor coefficients $A (n_1, n_2, n_3, n_4)$ of the rational function \\begin{equation*}\n  \\frac{1}{(1 - x_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0854","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:22:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CXtEcs5Mi4roj1Rx3Am6+vC4XwlT83xHmqaP7GQOVT86e60tOPCHs/IMRX269+qVR9aVE/0IcrWScxZqX2gYBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:15:52.474414Z"},"content_sha256":"497dd531b724db76e9c01976a4a7a1abc40e8e7cf406e1fc8a806958cf6d9638","schema_version":"1.0","event_id":"sha256:497dd531b724db76e9c01976a4a7a1abc40e8e7cf406e1fc8a806958cf6d9638"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GIW6H66WI75UPRZ5N2IVOGIY47/bundle.json","state_url":"https://pith.science/pith/GIW6H66WI75UPRZ5N2IVOGIY47/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GIW6H66WI75UPRZ5N2IVOGIY47/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":"2026-05-27T21:15:52Z","links":{"resolver":"https://pith.science/pith/GIW6H66WI75UPRZ5N2IVOGIY47","bundle":"https://pith.science/pith/GIW6H66WI75UPRZ5N2IVOGIY47/bundle.json","state":"https://pith.science/pith/GIW6H66WI75UPRZ5N2IVOGIY47/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GIW6H66WI75UPRZ5N2IVOGIY47/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GIW6H66WI75UPRZ5N2IVOGIY47","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":"8e30eff0999ff94693ad0c32461b7f5f09c9c51ed9db471886acec56b95f293d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-05T00:21:36Z","title_canon_sha256":"759253de15e3fa1220aae95dab31102bc8bc1b4dea77cee2bb93da12da88b9a6"},"schema_version":"1.0","source":{"id":"1401.0854","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.0854","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"arxiv_version","alias_value":"1401.0854v2","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.0854","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"pith_short_12","alias_value":"GIW6H66WI75U","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GIW6H66WI75UPRZ5","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GIW6H66W","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:497dd531b724db76e9c01976a4a7a1abc40e8e7cf406e1fc8a806958cf6d9638","target":"graph","created_at":"2026-05-18T01:22:24Z","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"},"paper":{"abstract_excerpt":"One of the many remarkable properties of the Ap\\'ery numbers $A (n)$, introduced in Ap\\'ery's proof of the irrationality of $\\zeta (3)$, is that they satisfy the two-term supercongruences \\begin{equation*}\n  A (p^r m) \\equiv A (p^{r - 1} m) \\pmod{p^{3 r}} \\end{equation*} for primes $p \\geq 5$. Similar congruences are conjectured to hold for all Ap\\'ery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Ap\\'ery numbers by showing that they extend to all Taylor coefficients $A (n_1, n_2, n_3, n_4)$ of the rational function \\begin{equation*}\n  \\frac{1}{(1 - x_","authors_text":"Armin Straub","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-05T00:21:36Z","title":"Multivariate Ap\\'ery numbers and supercongruences of rational functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0854","kind":"arxiv","version":2},"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":"sha256:6260b971d001e132f220f6fc3770d565a89342621161efce9a96f355ef69975c","target":"record","created_at":"2026-05-18T01:22:24Z","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":"8e30eff0999ff94693ad0c32461b7f5f09c9c51ed9db471886acec56b95f293d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-05T00:21:36Z","title_canon_sha256":"759253de15e3fa1220aae95dab31102bc8bc1b4dea77cee2bb93da12da88b9a6"},"schema_version":"1.0","source":{"id":"1401.0854","kind":"arxiv","version":2}},"canonical_sha256":"322de3fbd647fb47c73d6e91571918e7e7dd648edaaa0dbce19d3f8163911f12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"322de3fbd647fb47c73d6e91571918e7e7dd648edaaa0dbce19d3f8163911f12","first_computed_at":"2026-05-18T01:22:24.529876Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:24.529876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GA2E2h+MWElvHxqdfFrqxZRU2QdTqYL6jAA/KUsyKrBSOmSK1U36bkyQI+CkRCACYWZGdwu4XjTkOlhV0XIzBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:24.530553Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.0854","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6260b971d001e132f220f6fc3770d565a89342621161efce9a96f355ef69975c","sha256:497dd531b724db76e9c01976a4a7a1abc40e8e7cf406e1fc8a806958cf6d9638"],"state_sha256":"815c992f99fb1d5c4a007ba09992ca0946a2fb4c4439abf16864d456158e8f85"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yvU0BzEsnytX0tUIR/2QYE6ffR/OrFiAMiMMQCcQRU6u2KPGVgGJYs/fHOej1VumscvBHnq2ba5I8IQqeq3ZAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T21:15:52.477800Z","bundle_sha256":"2553f6bbabeb9658bd95bfaee519f30e23f9e1f5bfcd4ab35efa31805462aea8"}}