{"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"}