{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BZ4DFNDY7Q54GMHEHRRJDUTIR2","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":"930191cde30ba33b1ac4b4914d9e683e0657041f0fd50044037c7c12c1624ea6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-20T17:34:21Z","title_canon_sha256":"42f5995e91b1653dfec1be8f55bd38fdca9aebc5eb59277ee339f6abb628d847"},"schema_version":"1.0","source":{"id":"1701.05868","kind":"arxiv","version":10}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.05868","created_at":"2026-07-05T00:11:14Z"},{"alias_kind":"arxiv_version","alias_value":"1701.05868v10","created_at":"2026-07-05T00:11:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.05868","created_at":"2026-07-05T00:11:14Z"},{"alias_kind":"pith_short_12","alias_value":"BZ4DFNDY7Q54","created_at":"2026-07-05T00:11:14Z"},{"alias_kind":"pith_short_16","alias_value":"BZ4DFNDY7Q54GMHE","created_at":"2026-07-05T00:11:14Z"},{"alias_kind":"pith_short_8","alias_value":"BZ4DFNDY","created_at":"2026-07-05T00:11:14Z"}],"graph_snapshots":[{"event_id":"sha256:2eacae24ee6fff8106a7d97b788ec2469d1cb8edf7538df2c7c10709bebff685","target":"graph","created_at":"2026-07-05T00:11:14Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1701.05868/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\\in\\mathbb N=\\{0,1,2,\\ldots\\})$ with $|x+y-z|\\in\\{4^k:\\ k\\in\\mathbb N\\}$ (or $|2x-y|\\in\\{4^k:\\ k\\in\\mathbb N\\}$, or $x+y-z\\in\\{\\pm 8^k:\\ k\\in\\mathbb N\\}\\cup\\{0\\}\\subseteq\\{t^3:\\ t\\in\\mathbb Z\\}$), and that we can write any positive integer as $x^2+y^2+z^2+w^2$ $(x,y,z,w\\in\\mathbb Z)$ with $x+y+2z$ (or $x+2y+2z$) a power of four. We also prove that any $n\\in\\mathbb N$ can be writte","authors_text":"Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-20T17:34:21Z","title":"Restricted sums of four squares"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05868","kind":"arxiv","version":10},"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":"sha256:60c59fcc5314169d0ff78475a4123acf6a5f6922b926e89866dba26839efaa83","target":"record","created_at":"2026-07-05T00:11:14Z","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":"930191cde30ba33b1ac4b4914d9e683e0657041f0fd50044037c7c12c1624ea6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-20T17:34:21Z","title_canon_sha256":"42f5995e91b1653dfec1be8f55bd38fdca9aebc5eb59277ee339f6abb628d847"},"schema_version":"1.0","source":{"id":"1701.05868","kind":"arxiv","version":10}},"canonical_sha256":"0e7832b478fc3bc330e43c6291d2688e968b26e16e8b80ec76f00f1405262ae8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e7832b478fc3bc330e43c6291d2688e968b26e16e8b80ec76f00f1405262ae8","first_computed_at":"2026-07-05T00:11:14.327044Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:11:14.327044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fmw+pDD9SRH+dQfkBcEdU6v9Zq+Ew3w4Pn9rfWcyfvKaCPqHeBJ22WmhpyM1Td6wjFU9sAaVF2TzwI9XoJgpCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:11:14.327426Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.05868","source_kind":"arxiv","source_version":10}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:60c59fcc5314169d0ff78475a4123acf6a5f6922b926e89866dba26839efaa83","sha256:2eacae24ee6fff8106a7d97b788ec2469d1cb8edf7538df2c7c10709bebff685"],"state_sha256":"d7f4a12feceecaaa125617c5accbaec2c8239b8115ed1dd3f96d85ef0cac4d1c"}