{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:BD6MYHZPEEBA3FOY6RSGFHFIGJ","short_pith_number":"pith:BD6MYHZP","canonical_record":{"source":{"id":"1502.00366","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-02T05:47:16Z","cross_cats_sorted":[],"title_canon_sha256":"ca8f4aca2eded2afbbb4343633aa5e2d128382bea422cc0f4b97d3a6ba324302","abstract_canon_sha256":"55b779246e5eb0b638aaa051dfc9adf725b5f076066351929cb8c93074cf84e8"},"schema_version":"1.0"},"canonical_sha256":"08fccc1f2f21020d95d8f464629ca83261462dc3f73ec584dc952edb801ba2cd","source":{"kind":"arxiv","id":"1502.00366","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.00366","created_at":"2026-05-18T01:15:38Z"},{"alias_kind":"arxiv_version","alias_value":"1502.00366v4","created_at":"2026-05-18T01:15:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00366","created_at":"2026-05-18T01:15:38Z"},{"alias_kind":"pith_short_12","alias_value":"BD6MYHZPEEBA","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BD6MYHZPEEBA3FOY","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BD6MYHZP","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:BD6MYHZPEEBA3FOY6RSGFHFIGJ","target":"record","payload":{"canonical_record":{"source":{"id":"1502.00366","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-02T05:47:16Z","cross_cats_sorted":[],"title_canon_sha256":"ca8f4aca2eded2afbbb4343633aa5e2d128382bea422cc0f4b97d3a6ba324302","abstract_canon_sha256":"55b779246e5eb0b638aaa051dfc9adf725b5f076066351929cb8c93074cf84e8"},"schema_version":"1.0"},"canonical_sha256":"08fccc1f2f21020d95d8f464629ca83261462dc3f73ec584dc952edb801ba2cd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:38.815622Z","signature_b64":"3/+wxRYomNRqnFUm9C1eR3zrWR8SK6INY4SHrX7BhB17ZHjYG7biWAb2GYA3BZ693iR2GKDQHpQxq1zDUR+DDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08fccc1f2f21020d95d8f464629ca83261462dc3f73ec584dc952edb801ba2cd","last_reissued_at":"2026-05-18T01:15:38.814466Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:38.814466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.00366","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:15:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GdmMZbnpozFn7OQzqK7DGMG75iR6axxVNabjFfZfIhGXiftUAAq0BuaszIARPsuD5gVQeGKqtj5s6Q8v3Y6fBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:07:04.002790Z"},"content_sha256":"31bc8f8f8bc2570934d973482258118764b00759bd7c7757cdc682a91819a87e","schema_version":"1.0","event_id":"sha256:31bc8f8f8bc2570934d973482258118764b00759bd7c7757cdc682a91819a87e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:BD6MYHZPEEBA3FOY6RSGFHFIGJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Partitions into a small number of part sizes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"William J. Keith","submitted_at":"2015-02-02T05:47:16Z","abstract_excerpt":"We study $\\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\\nu_2$ and $\\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\\nu_2(An+B) \\equiv 0 \\pmod{4}$ for (A,B) = (36,30), (72,42), (252,114), (196,70), and likely many other progressions for which our method should easily generalize. Of some independent interest, we prove that the overpartition function $\\bar{p}(n) \\equiv 0 \\pmod{16}$ in the first three progressions (the fourth is known), and thereby show that $\\nu_3(An+B) \\equiv 0 \\pmod{2}$ in each of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00366","kind":"arxiv","version":4},"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":"2026-05-18T01:15:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h6DFRnGm4LutjKwDX7nh36yPXXMoRn4+LbXmj4s3ELpgUemBvleJhi6hco250+6yM7/iMH/zftf1G+atXojRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:07:04.003425Z"},"content_sha256":"693bd9daa1b90f7d474fd2287dc0f1825dc6ac7e22ea9ee87219e2aff36f1a61","schema_version":"1.0","event_id":"sha256:693bd9daa1b90f7d474fd2287dc0f1825dc6ac7e22ea9ee87219e2aff36f1a61"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BD6MYHZPEEBA3FOY6RSGFHFIGJ/bundle.json","state_url":"https://pith.science/pith/BD6MYHZPEEBA3FOY6RSGFHFIGJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BD6MYHZPEEBA3FOY6RSGFHFIGJ/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":"2026-06-06T20:07:04Z","links":{"resolver":"https://pith.science/pith/BD6MYHZPEEBA3FOY6RSGFHFIGJ","bundle":"https://pith.science/pith/BD6MYHZPEEBA3FOY6RSGFHFIGJ/bundle.json","state":"https://pith.science/pith/BD6MYHZPEEBA3FOY6RSGFHFIGJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BD6MYHZPEEBA3FOY6RSGFHFIGJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BD6MYHZPEEBA3FOY6RSGFHFIGJ","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":"55b779246e5eb0b638aaa051dfc9adf725b5f076066351929cb8c93074cf84e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-02T05:47:16Z","title_canon_sha256":"ca8f4aca2eded2afbbb4343633aa5e2d128382bea422cc0f4b97d3a6ba324302"},"schema_version":"1.0","source":{"id":"1502.00366","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.00366","created_at":"2026-05-18T01:15:38Z"},{"alias_kind":"arxiv_version","alias_value":"1502.00366v4","created_at":"2026-05-18T01:15:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00366","created_at":"2026-05-18T01:15:38Z"},{"alias_kind":"pith_short_12","alias_value":"BD6MYHZPEEBA","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BD6MYHZPEEBA3FOY","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BD6MYHZP","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:693bd9daa1b90f7d474fd2287dc0f1825dc6ac7e22ea9ee87219e2aff36f1a61","target":"graph","created_at":"2026-05-18T01:15:38Z","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":"We study $\\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\\nu_2$ and $\\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\\nu_2(An+B) \\equiv 0 \\pmod{4}$ for (A,B) = (36,30), (72,42), (252,114), (196,70), and likely many other progressions for which our method should easily generalize. Of some independent interest, we prove that the overpartition function $\\bar{p}(n) \\equiv 0 \\pmod{16}$ in the first three progressions (the fourth is known), and thereby show that $\\nu_3(An+B) \\equiv 0 \\pmod{2}$ in each of t","authors_text":"William J. Keith","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-02T05:47:16Z","title":"Partitions into a small number of part sizes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00366","kind":"arxiv","version":4},"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":"sha256:31bc8f8f8bc2570934d973482258118764b00759bd7c7757cdc682a91819a87e","target":"record","created_at":"2026-05-18T01:15:38Z","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":"55b779246e5eb0b638aaa051dfc9adf725b5f076066351929cb8c93074cf84e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-02T05:47:16Z","title_canon_sha256":"ca8f4aca2eded2afbbb4343633aa5e2d128382bea422cc0f4b97d3a6ba324302"},"schema_version":"1.0","source":{"id":"1502.00366","kind":"arxiv","version":4}},"canonical_sha256":"08fccc1f2f21020d95d8f464629ca83261462dc3f73ec584dc952edb801ba2cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"08fccc1f2f21020d95d8f464629ca83261462dc3f73ec584dc952edb801ba2cd","first_computed_at":"2026-05-18T01:15:38.814466Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:38.814466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3/+wxRYomNRqnFUm9C1eR3zrWR8SK6INY4SHrX7BhB17ZHjYG7biWAb2GYA3BZ693iR2GKDQHpQxq1zDUR+DDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:38.815622Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.00366","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31bc8f8f8bc2570934d973482258118764b00759bd7c7757cdc682a91819a87e","sha256:693bd9daa1b90f7d474fd2287dc0f1825dc6ac7e22ea9ee87219e2aff36f1a61"],"state_sha256":"0e3409f94a761f376f75bfe07e63c0dd795148bf0913c45e1df34520689ee54d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BDp0jzrSzxUSBNGjn3K+x4NjCMGqQ3GGAjNMb6vfNqyOGKSl+ZShd332EGfnMAs+Z+1lq/x11qnXtkY+BQobDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:07:04.006213Z","bundle_sha256":"bce59d482367e0c0d196fe338532094affc8da22d8fe3b4659a8a69b1841b39c"}}