{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:MVUZDZN75B6EZQLHZVIDVNLREI","short_pith_number":"pith:MVUZDZN7","canonical_record":{"source":{"id":"1907.11902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-27T12:11:27Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"653789306a8d386e1508b87669f59ac80a63147e136e2d39a86be16e104e2a83","abstract_canon_sha256":"48648db14afe66ad6b897ac07d4bc2abcd588007031b3184a67a18f5359e23d3"},"schema_version":"1.0"},"canonical_sha256":"656991e5bfe87c4cc167cd503ab571222545ef2479c1f70c8e610ec86293611b","source":{"kind":"arxiv","id":"1907.11902","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.11902","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"arxiv_version","alias_value":"1907.11902v1","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.11902","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"pith_short_12","alias_value":"MVUZDZN75B6E","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"MVUZDZN75B6EZQLH","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"MVUZDZN7","created_at":"2026-05-18T12:33:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:MVUZDZN75B6EZQLHZVIDVNLREI","target":"record","payload":{"canonical_record":{"source":{"id":"1907.11902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-27T12:11:27Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"653789306a8d386e1508b87669f59ac80a63147e136e2d39a86be16e104e2a83","abstract_canon_sha256":"48648db14afe66ad6b897ac07d4bc2abcd588007031b3184a67a18f5359e23d3"},"schema_version":"1.0"},"canonical_sha256":"656991e5bfe87c4cc167cd503ab571222545ef2479c1f70c8e610ec86293611b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:22.589880Z","signature_b64":"av+avPzmBjKFBxKqAhEw8ChBmPa3+GwyReejdpSccf9dk27/5w6U+yY3Ov0yDowRNg0vR/Nn36er6UsttXM3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"656991e5bfe87c4cc167cd503ab571222545ef2479c1f70c8e610ec86293611b","last_reissued_at":"2026-05-17T23:39:22.589257Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:22.589257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.11902","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HAY49ikrpijh+OWWLbWV9hgEwlvFE+501SLa4/Fp70HmzGuY6HBAfenUWuzfrGHAgt6+R33EF9SaVt08X1cMCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:19:52.111678Z"},"content_sha256":"40f298a4c21a033e728c927fd92f555d20dbd7a26959aa028acecf88ad8e2a71","schema_version":"1.0","event_id":"sha256:40f298a4c21a033e728c927fd92f555d20dbd7a26959aa028acecf88ad8e2a71"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:MVUZDZN75B6EZQLHZVIDVNLREI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Legendre's formula and $p$-adic analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Gennady Eremin","submitted_at":"2019-07-27T12:11:27Z","abstract_excerpt":"In number theory, we know Legendre's formula $ v_p(n!) = \\sum_{k \\ge 1} \\lfloor \\frac{n}{p^k} \\rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $ v_p (n!) = \\frac{n-s_p(n)}{p-1} $ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The artic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.11902","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":""},"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":"2026-05-17T23:39:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gndb0XUL5/FtyauB7HUfBU6lwQ5mJtzFPwLT6yNOiCcYSjBWYLpbVqX7unQIuPX/Wz6jevv7CWR4aqcqjRXADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:19:52.112308Z"},"content_sha256":"5dc2727273c65ef0fa8efa3ddcc0b09f9194c285b100cdbf1e52d00270875c8a","schema_version":"1.0","event_id":"sha256:5dc2727273c65ef0fa8efa3ddcc0b09f9194c285b100cdbf1e52d00270875c8a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MVUZDZN75B6EZQLHZVIDVNLREI/bundle.json","state_url":"https://pith.science/pith/MVUZDZN75B6EZQLHZVIDVNLREI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MVUZDZN75B6EZQLHZVIDVNLREI/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":"2026-05-26T18:19:52Z","links":{"resolver":"https://pith.science/pith/MVUZDZN75B6EZQLHZVIDVNLREI","bundle":"https://pith.science/pith/MVUZDZN75B6EZQLHZVIDVNLREI/bundle.json","state":"https://pith.science/pith/MVUZDZN75B6EZQLHZVIDVNLREI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MVUZDZN75B6EZQLHZVIDVNLREI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:MVUZDZN75B6EZQLHZVIDVNLREI","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":"48648db14afe66ad6b897ac07d4bc2abcd588007031b3184a67a18f5359e23d3","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-27T12:11:27Z","title_canon_sha256":"653789306a8d386e1508b87669f59ac80a63147e136e2d39a86be16e104e2a83"},"schema_version":"1.0","source":{"id":"1907.11902","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.11902","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"arxiv_version","alias_value":"1907.11902v1","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.11902","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"pith_short_12","alias_value":"MVUZDZN75B6E","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"MVUZDZN75B6EZQLH","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"MVUZDZN7","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:5dc2727273c65ef0fa8efa3ddcc0b09f9194c285b100cdbf1e52d00270875c8a","target":"graph","created_at":"2026-05-17T23:39:22Z","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":"In number theory, we know Legendre's formula $ v_p(n!) = \\sum_{k \\ge 1} \\lfloor \\frac{n}{p^k} \\rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $ v_p (n!) = \\frac{n-s_p(n)}{p-1} $ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The artic","authors_text":"Gennady Eremin","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-27T12:11:27Z","title":"Legendre's formula and $p$-adic analysis"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.11902","kind":"arxiv","version":1},"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":"sha256:40f298a4c21a033e728c927fd92f555d20dbd7a26959aa028acecf88ad8e2a71","target":"record","created_at":"2026-05-17T23:39:22Z","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":"48648db14afe66ad6b897ac07d4bc2abcd588007031b3184a67a18f5359e23d3","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-27T12:11:27Z","title_canon_sha256":"653789306a8d386e1508b87669f59ac80a63147e136e2d39a86be16e104e2a83"},"schema_version":"1.0","source":{"id":"1907.11902","kind":"arxiv","version":1}},"canonical_sha256":"656991e5bfe87c4cc167cd503ab571222545ef2479c1f70c8e610ec86293611b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"656991e5bfe87c4cc167cd503ab571222545ef2479c1f70c8e610ec86293611b","first_computed_at":"2026-05-17T23:39:22.589257Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:22.589257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"av+avPzmBjKFBxKqAhEw8ChBmPa3+GwyReejdpSccf9dk27/5w6U+yY3Ov0yDowRNg0vR/Nn36er6UsttXM3Cw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:22.589880Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.11902","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40f298a4c21a033e728c927fd92f555d20dbd7a26959aa028acecf88ad8e2a71","sha256:5dc2727273c65ef0fa8efa3ddcc0b09f9194c285b100cdbf1e52d00270875c8a"],"state_sha256":"696a6f7ca391d020db7b7748488c2faea2859cacd33925223e32a4dc5460c498"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9xyq4QlFoNvjL3mSX9RD6mvogeTV6XTFTYlcZt7HbGdUZD6Vufx9QoQloLEAHTHsd8ZSYrrtjQ7JeNSk3r9rAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T18:19:52.115789Z","bundle_sha256":"af4b76b8153f052c4dbec26b46ed94c23748b8ecc6d8210ae68a2b7add2e0c10"}}