{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:B2ZGVRTMAMOFUZXR4YK2TVGBKH","short_pith_number":"pith:B2ZGVRTM","canonical_record":{"source":{"id":"1111.0572","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-02T17:30:10Z","cross_cats_sorted":[],"title_canon_sha256":"6dcf5c068b4dd3b236e8fa892363ab16452988f9fa549ac9b79ed54b37f06507","abstract_canon_sha256":"314054391d0672f382bf876912b99313f742e1535b47a76a71904f8d43fe6480"},"schema_version":"1.0"},"canonical_sha256":"0eb26ac66c031c5a66f1e615a9d4c151d01b6a25edfd916ec27681ad733d7e3f","source":{"kind":"arxiv","id":"1111.0572","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.0572","created_at":"2026-05-18T04:09:44Z"},{"alias_kind":"arxiv_version","alias_value":"1111.0572v1","created_at":"2026-05-18T04:09:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.0572","created_at":"2026-05-18T04:09:44Z"},{"alias_kind":"pith_short_12","alias_value":"B2ZGVRTMAMOF","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"B2ZGVRTMAMOFUZXR","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"B2ZGVRTM","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:B2ZGVRTMAMOFUZXR4YK2TVGBKH","target":"record","payload":{"canonical_record":{"source":{"id":"1111.0572","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-02T17:30:10Z","cross_cats_sorted":[],"title_canon_sha256":"6dcf5c068b4dd3b236e8fa892363ab16452988f9fa549ac9b79ed54b37f06507","abstract_canon_sha256":"314054391d0672f382bf876912b99313f742e1535b47a76a71904f8d43fe6480"},"schema_version":"1.0"},"canonical_sha256":"0eb26ac66c031c5a66f1e615a9d4c151d01b6a25edfd916ec27681ad733d7e3f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:44.661742Z","signature_b64":"rprl2J132q7RTvuADpaqX/XYxraQksK2hJyvQDBWlNKwEBCvkuq4SgvB6Ta8Zq8Mbn/Blf42KnQBnwyt4DF7CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0eb26ac66c031c5a66f1e615a9d4c151d01b6a25edfd916ec27681ad733d7e3f","last_reissued_at":"2026-05-18T04:09:44.661184Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:44.661184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.0572","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":"2026-05-18T04:09:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1fnybFf43l6JLdFMxHsaNwmydU2txsUpG8t6Fa4UOXiQmoVL6F+8sNU8cKKYzXbjd+4kUHODmkLZK3uYK9oPDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:14:56.016784Z"},"content_sha256":"478ef654f68a2463025f05efcffbc200c656769a016fecc426e809be7b8483fd","schema_version":"1.0","event_id":"sha256:478ef654f68a2463025f05efcffbc200c656769a016fecc426e809be7b8483fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:B2ZGVRTMAMOFUZXR4YK2TVGBKH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finding elementary formulas for theta functions associated to even sums of squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ila Varma","submitted_at":"2011-11-02T17:30:10Z","abstract_excerpt":"This article discusses the classical problem of how to calculate $r_n(m)$, the number of ways to represent an integer $m$ by a sum of $n$ squares from a computational efficiency viewpoint. Although this problem has been studied in great detail, there are very few formulas given for the purpose of computing $r_n(m)$ quickly. More precisely, for fixed $n$, we want a formula for $r_n(m)$ that computes in log-polynomial time (with respect to $m$) when the prime factorization of $m$ is given. Restricting to even $n$, we can view $\\theta_n(q)$, the theta function associated to sums of $n$ squares, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0572","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":"2026-05-18T04:09:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IEI1JfN9FP1SYqbpquZlLoppPSb5Y6cSwV9ZfdN7M76QcmxAt6uifhKLVTOUVSJCXco6vcS0b104a4m/amfoCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:14:56.017507Z"},"content_sha256":"8065059c762f85c43bb6dfd0a7b97446d466b18be5d0cb4ecaa8e292e0b11e19","schema_version":"1.0","event_id":"sha256:8065059c762f85c43bb6dfd0a7b97446d466b18be5d0cb4ecaa8e292e0b11e19"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B2ZGVRTMAMOFUZXR4YK2TVGBKH/bundle.json","state_url":"https://pith.science/pith/B2ZGVRTMAMOFUZXR4YK2TVGBKH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B2ZGVRTMAMOFUZXR4YK2TVGBKH/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":"2026-05-31T03:14:56Z","links":{"resolver":"https://pith.science/pith/B2ZGVRTMAMOFUZXR4YK2TVGBKH","bundle":"https://pith.science/pith/B2ZGVRTMAMOFUZXR4YK2TVGBKH/bundle.json","state":"https://pith.science/pith/B2ZGVRTMAMOFUZXR4YK2TVGBKH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B2ZGVRTMAMOFUZXR4YK2TVGBKH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:B2ZGVRTMAMOFUZXR4YK2TVGBKH","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":"314054391d0672f382bf876912b99313f742e1535b47a76a71904f8d43fe6480","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-02T17:30:10Z","title_canon_sha256":"6dcf5c068b4dd3b236e8fa892363ab16452988f9fa549ac9b79ed54b37f06507"},"schema_version":"1.0","source":{"id":"1111.0572","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.0572","created_at":"2026-05-18T04:09:44Z"},{"alias_kind":"arxiv_version","alias_value":"1111.0572v1","created_at":"2026-05-18T04:09:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.0572","created_at":"2026-05-18T04:09:44Z"},{"alias_kind":"pith_short_12","alias_value":"B2ZGVRTMAMOF","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"B2ZGVRTMAMOFUZXR","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"B2ZGVRTM","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:8065059c762f85c43bb6dfd0a7b97446d466b18be5d0cb4ecaa8e292e0b11e19","target":"graph","created_at":"2026-05-18T04:09:44Z","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":"This article discusses the classical problem of how to calculate $r_n(m)$, the number of ways to represent an integer $m$ by a sum of $n$ squares from a computational efficiency viewpoint. Although this problem has been studied in great detail, there are very few formulas given for the purpose of computing $r_n(m)$ quickly. More precisely, for fixed $n$, we want a formula for $r_n(m)$ that computes in log-polynomial time (with respect to $m$) when the prime factorization of $m$ is given. Restricting to even $n$, we can view $\\theta_n(q)$, the theta function associated to sums of $n$ squares, a","authors_text":"Ila Varma","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-02T17:30:10Z","title":"Finding elementary formulas for theta functions associated to even sums of squares"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0572","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":"sha256:478ef654f68a2463025f05efcffbc200c656769a016fecc426e809be7b8483fd","target":"record","created_at":"2026-05-18T04:09:44Z","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":"314054391d0672f382bf876912b99313f742e1535b47a76a71904f8d43fe6480","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-02T17:30:10Z","title_canon_sha256":"6dcf5c068b4dd3b236e8fa892363ab16452988f9fa549ac9b79ed54b37f06507"},"schema_version":"1.0","source":{"id":"1111.0572","kind":"arxiv","version":1}},"canonical_sha256":"0eb26ac66c031c5a66f1e615a9d4c151d01b6a25edfd916ec27681ad733d7e3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0eb26ac66c031c5a66f1e615a9d4c151d01b6a25edfd916ec27681ad733d7e3f","first_computed_at":"2026-05-18T04:09:44.661184Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:44.661184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rprl2J132q7RTvuADpaqX/XYxraQksK2hJyvQDBWlNKwEBCvkuq4SgvB6Ta8Zq8Mbn/Blf42KnQBnwyt4DF7CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:44.661742Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.0572","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:478ef654f68a2463025f05efcffbc200c656769a016fecc426e809be7b8483fd","sha256:8065059c762f85c43bb6dfd0a7b97446d466b18be5d0cb4ecaa8e292e0b11e19"],"state_sha256":"f61e9af08548fb3dea3ea156d1e0b903ba912ec8dd89d7ec1c8115a74b10d7ba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zSab1Ytpbph8Qb+cx4UfCXyFGESW6YAXGDaMfR+N3FJn40Egvio8/Z+8rLtIfa9yocpTnuzdOqrFAVdGqNz2CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T03:14:56.021261Z","bundle_sha256":"7a16ed22c9af92ecbf93728ae18df25c80efdb8954b4c0be40df4fa672ec620a"}}