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Then every $C_{4m}$ is a sum of $C_{k}$ with $k<4m$.\n  This, combined with earlier results, yields: If $K$ consists of all mod $2$ modular forms of level $\\Gamma_{0}(3)$ annihilated by $U_{2}$ and $U_{3} +I$, then $K$ has a basis adapted (in the sense of Nicolas and Serre) to the Hecke operators $T_{7}$ and $T_{13}$; consequently the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1603.03910","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-12T13:26:41Z","cross_cats_sorted":[],"title_canon_sha256":"edd6ed9b0ead1b39f641b3391996c9199ecff0e9b67b9a8775895905546a245a","abstract_canon_sha256":"a045ac8fbfa21809b94910f6e42ddb4c4611cf48f95d4abb4c68ccf43273a100"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:10.071333Z","signature_b64":"wnAFbA9LgnsNdB2YPEccqtI8riR+CBZzm0vMqF9Timy/7DvfXyRKb7/OP2K4Jk5dCepZSYnzw1A7Hu6j6Lx1CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c02c97d14db9dff9c55b587a36e50e98e45c7f268748f0456aa43d90bde1806e","last_reissued_at":"2026-05-18T01:19:10.070661Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:10.070661Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characteristic 2 recurrence related to $U_3$, with a Hecke algebra application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paul Monsky","submitted_at":"2016-03-12T13:26:41Z","abstract_excerpt":"I begin with a simple modular form motivated proof of the following: Let $C_{n}$ in $Z/2[[t]]$ be defined by $C_{n+4} = C_{n+3} + (t^{4}+t^{3}+t^{2}+t)C_{n} + t^{n}(t^{2}+t)$, with initial values $0$, $1$, $t$ and $t^{2}$ for $C_{0}$, $C_{1}$, $C_{2}$ and $C_{3}$. Then every $C_{4m}$ is a sum of $C_{k}$ with $k<4m$.\n  This, combined with earlier results, yields: If $K$ consists of all mod $2$ modular forms of level $\\Gamma_{0}(3)$ annihilated by $U_{2}$ and $U_{3} +I$, then $K$ has a basis adapted (in the sense of Nicolas and Serre) to the Hecke operators $T_{7}$ and $T_{13}$; consequently the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03910","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.03910","created_at":"2026-05-18T01:19:10.070754+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.03910v1","created_at":"2026-05-18T01:19:10.070754+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.03910","created_at":"2026-05-18T01:19:10.070754+00:00"},{"alias_kind":"pith_short_12","alias_value":"YAWJPUKNXHP7","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"YAWJPUKNXHP7TRK3","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"YAWJPUKN","created_at":"2026-05-18T12:30:53.716459+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD","json":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD.json","graph_json":"https://pith.science/api/pith-number/YAWJPUKNXHP7TRK3LB5DNZIOTD/graph.json","events_json":"https://pith.science/api/pith-number/YAWJPUKNXHP7TRK3LB5DNZIOTD/events.json","paper":"https://pith.science/paper/YAWJPUKN"},"agent_actions":{"view_html":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD","download_json":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD.json","view_paper":"https://pith.science/paper/YAWJPUKN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.03910&json=true","fetch_graph":"https://pith.science/api/pith-number/YAWJPUKNXHP7TRK3LB5DNZIOTD/graph.json","fetch_events":"https://pith.science/api/pith-number/YAWJPUKNXHP7TRK3LB5DNZIOTD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD/action/storage_attestation","attest_author":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD/action/author_attestation","sign_citation":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD/action/citation_signature","submit_replication":"https://pith.science/pith/YAWJPUKNXHP7TRK3LB5DNZIOTD/action/replication_record"}},"created_at":"2026-05-18T01:19:10.070754+00:00","updated_at":"2026-05-18T01:19:10.070754+00:00"}