{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:67JQ4V4X5LRLHVAC6VILX4IMEV","short_pith_number":"pith:67JQ4V4X","canonical_record":{"source":{"id":"1805.03038","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T14:06:02Z","cross_cats_sorted":[],"title_canon_sha256":"648fd57ee4d5451276cddc46e8e968675ecb307e71b4a0ed7e1cb029f5d63ee3","abstract_canon_sha256":"ba12f802b4d6dbf0f3b1e3245d2ea02239ba2c1ca572df2c6937b9e3d3df2624"},"schema_version":"1.0"},"canonical_sha256":"f7d30e5797eae2b3d402f550bbf10c2571a9c772b54bbae9ac7d59ab1ba63d31","source":{"kind":"arxiv","id":"1805.03038","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.03038","created_at":"2026-05-18T00:16:34Z"},{"alias_kind":"arxiv_version","alias_value":"1805.03038v1","created_at":"2026-05-18T00:16:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.03038","created_at":"2026-05-18T00:16:34Z"},{"alias_kind":"pith_short_12","alias_value":"67JQ4V4X5LRL","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"67JQ4V4X5LRLHVAC","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"67JQ4V4X","created_at":"2026-05-18T12:32:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:67JQ4V4X5LRLHVAC6VILX4IMEV","target":"record","payload":{"canonical_record":{"source":{"id":"1805.03038","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T14:06:02Z","cross_cats_sorted":[],"title_canon_sha256":"648fd57ee4d5451276cddc46e8e968675ecb307e71b4a0ed7e1cb029f5d63ee3","abstract_canon_sha256":"ba12f802b4d6dbf0f3b1e3245d2ea02239ba2c1ca572df2c6937b9e3d3df2624"},"schema_version":"1.0"},"canonical_sha256":"f7d30e5797eae2b3d402f550bbf10c2571a9c772b54bbae9ac7d59ab1ba63d31","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:34.828625Z","signature_b64":"giOmSUhQgQJYhioqjoflrBpFEylXrunmagbFIL7Z9Yp3KBR3OP/ol7nYtP5ajczz3tW+bkR7mbZWcvX0yC4HAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7d30e5797eae2b3d402f550bbf10c2571a9c772b54bbae9ac7d59ab1ba63d31","last_reissued_at":"2026-05-18T00:16:34.828006Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:34.828006Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.03038","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":"2026-05-18T00:16:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nLvrlmVRkx/aJ/Ct/b8imjQvVmx+boo7n0xDCz2VIDSHHDEiShzi1bD3afqgilX3xUFLc0rQwRwVKR3u+88TCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T12:30:51.016368Z"},"content_sha256":"aa1cada3dc5502629a9ef2c183a9fdf7ffb4b7affbf1738200a71423132e933e","schema_version":"1.0","event_id":"sha256:aa1cada3dc5502629a9ef2c183a9fdf7ffb4b7affbf1738200a71423132e933e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:67JQ4V4X5LRLHVAC6VILX4IMEV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A sum of squares not divisible by a prime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Byeong-Kweon Oh, Kyoungmin Kim","submitted_at":"2018-05-08T14:06:02Z","abstract_excerpt":"Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$, except the unique positive integer $79$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03038","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":"2026-05-18T00:16:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xsB9cBm25YjM7KtQTfgGY/PF7qpAJt1wJFLQ3mgO8kc1xQTVn1uP62ZUBe/n/nn1wR/eG2uWizAt6QvCL5pNCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T12:30:51.016745Z"},"content_sha256":"138cd885615f71fe3c1a66ee6284ae5b5a8d3ca4082cd7b4e79765f464cf1bc2","schema_version":"1.0","event_id":"sha256:138cd885615f71fe3c1a66ee6284ae5b5a8d3ca4082cd7b4e79765f464cf1bc2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/67JQ4V4X5LRLHVAC6VILX4IMEV/bundle.json","state_url":"https://pith.science/pith/67JQ4V4X5LRLHVAC6VILX4IMEV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/67JQ4V4X5LRLHVAC6VILX4IMEV/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":"2026-06-05T12:30:51Z","links":{"resolver":"https://pith.science/pith/67JQ4V4X5LRLHVAC6VILX4IMEV","bundle":"https://pith.science/pith/67JQ4V4X5LRLHVAC6VILX4IMEV/bundle.json","state":"https://pith.science/pith/67JQ4V4X5LRLHVAC6VILX4IMEV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/67JQ4V4X5LRLHVAC6VILX4IMEV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:67JQ4V4X5LRLHVAC6VILX4IMEV","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":"ba12f802b4d6dbf0f3b1e3245d2ea02239ba2c1ca572df2c6937b9e3d3df2624","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T14:06:02Z","title_canon_sha256":"648fd57ee4d5451276cddc46e8e968675ecb307e71b4a0ed7e1cb029f5d63ee3"},"schema_version":"1.0","source":{"id":"1805.03038","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.03038","created_at":"2026-05-18T00:16:34Z"},{"alias_kind":"arxiv_version","alias_value":"1805.03038v1","created_at":"2026-05-18T00:16:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.03038","created_at":"2026-05-18T00:16:34Z"},{"alias_kind":"pith_short_12","alias_value":"67JQ4V4X5LRL","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"67JQ4V4X5LRLHVAC","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"67JQ4V4X","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:138cd885615f71fe3c1a66ee6284ae5b5a8d3ca4082cd7b4e79765f464cf1bc2","target":"graph","created_at":"2026-05-18T00:16:34Z","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":"Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$, except the unique positive integer $79$.","authors_text":"Byeong-Kweon Oh, Kyoungmin Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T14:06:02Z","title":"A sum of squares not divisible by a prime"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03038","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:aa1cada3dc5502629a9ef2c183a9fdf7ffb4b7affbf1738200a71423132e933e","target":"record","created_at":"2026-05-18T00:16:34Z","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":"ba12f802b4d6dbf0f3b1e3245d2ea02239ba2c1ca572df2c6937b9e3d3df2624","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T14:06:02Z","title_canon_sha256":"648fd57ee4d5451276cddc46e8e968675ecb307e71b4a0ed7e1cb029f5d63ee3"},"schema_version":"1.0","source":{"id":"1805.03038","kind":"arxiv","version":1}},"canonical_sha256":"f7d30e5797eae2b3d402f550bbf10c2571a9c772b54bbae9ac7d59ab1ba63d31","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f7d30e5797eae2b3d402f550bbf10c2571a9c772b54bbae9ac7d59ab1ba63d31","first_computed_at":"2026-05-18T00:16:34.828006Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:34.828006Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"giOmSUhQgQJYhioqjoflrBpFEylXrunmagbFIL7Z9Yp3KBR3OP/ol7nYtP5ajczz3tW+bkR7mbZWcvX0yC4HAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:34.828625Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.03038","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa1cada3dc5502629a9ef2c183a9fdf7ffb4b7affbf1738200a71423132e933e","sha256:138cd885615f71fe3c1a66ee6284ae5b5a8d3ca4082cd7b4e79765f464cf1bc2"],"state_sha256":"d5851328c04fd1703a4073eda05403b11ad3f2c8d0201ae281a70e46b7e587ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6KYXcoVgz3QL/C2L36is4ZBzgi4g1DFf/pyepvLLykMdNRBdn6NrSNAVlpwRIK8y93Ax/fQUfm9tB4z2dORwDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T12:30:51.018684Z","bundle_sha256":"a31df35dac7162fc265ee26ae9a97445781e73fd9e73913e37b2caa99d7157d5"}}