{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NCWYHB5KPIKQAS3G4WFKKFIJSM","short_pith_number":"pith:NCWYHB5K","canonical_record":{"source":{"id":"2605.18644","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-18T16:51:26Z","cross_cats_sorted":[],"title_canon_sha256":"820725a3250158ed1828cd4aaba096d5196ca7138837ed4e792031e1c96802bd","abstract_canon_sha256":"be8371a20069eacc8f71e5ccebd9f06d9f699e6411dff3dc41f66f2133e77dcf"},"schema_version":"1.0"},"canonical_sha256":"68ad8387aa7a15004b66e58aa51509933175b3369348f9003f90919e4d4fcd94","source":{"kind":"arxiv","id":"2605.18644","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.18644","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"arxiv_version","alias_value":"2605.18644v1","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.18644","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"pith_short_12","alias_value":"NCWYHB5KPIKQ","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"pith_short_16","alias_value":"NCWYHB5KPIKQAS3G","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"pith_short_8","alias_value":"NCWYHB5K","created_at":"2026-05-20T00:06:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NCWYHB5KPIKQAS3G4WFKKFIJSM","target":"record","payload":{"canonical_record":{"source":{"id":"2605.18644","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-18T16:51:26Z","cross_cats_sorted":[],"title_canon_sha256":"820725a3250158ed1828cd4aaba096d5196ca7138837ed4e792031e1c96802bd","abstract_canon_sha256":"be8371a20069eacc8f71e5ccebd9f06d9f699e6411dff3dc41f66f2133e77dcf"},"schema_version":"1.0"},"canonical_sha256":"68ad8387aa7a15004b66e58aa51509933175b3369348f9003f90919e4d4fcd94","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:06:12.537502Z","signature_b64":"Yo9gLKBs1RcCOofJawjctbuvDQmZgw8AAEz5McGmQswexG5XPoy5cOPVBoI40iSwLrzn2ZQ69OcnaFdx/NN+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68ad8387aa7a15004b66e58aa51509933175b3369348f9003f90919e4d4fcd94","last_reissued_at":"2026-05-20T00:06:12.536703Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:06:12.536703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.18644","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":"2026-05-20T00:06:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hbxMOI04ujl/AvLV6gkEXVDmesLI0y/vAgFwSBlADC7Zzoz498VmXaDrkpDONTjdJIpWUASxLjA8u0cJYHG7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:10:54.710694Z"},"content_sha256":"b674fb1f47972d529feab7065e5659d6961b82e65db80e9df33b4f31cc0692b6","schema_version":"1.0","event_id":"sha256:b674fb1f47972d529feab7065e5659d6961b82e65db80e9df33b4f31cc0692b6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NCWYHB5KPIKQAS3G4WFKKFIJSM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Covering systems where the prime divisors of all moduli are only $2$, $3$, or $5$","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jonah Klein, Joshua Harrington, Joshua Lowrance, Ognian Trifonov","submitted_at":"2026-05-18T16:51:26Z","abstract_excerpt":"We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \\geq b \\geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete description of all such quadruples when $m=2,3,4,5$, or $6$, except when $m=6$ and $b=c=1$. We also show that if the LCM of the moduli has only $2$, $3$, or $5$ as prime divisors, then $m \\leq 9$ and construct a distinct covering system with $m=8$, $a=8$, $b=3$, and $c=2$. When a covering system exists for a quadruple $(m,a,b,c)$ we provide an example. Nonexisten"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.18644","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18644/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T00:01:59.181012Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"ac61f50557c1259a0687e5bb151feb71301f66915f4bc11434dbe7e70bea9d81"},"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":"2026-05-20T00:06:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Egu+eWbbz6PFi1CBq46DNchdX969kfe/XBseVIn+slA+ahekIYFy6VD/xwQePzKcyUtYGJtLZBMXntig+qJADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:10:54.711532Z"},"content_sha256":"dae5c3d00db4d39984bec0ff8f078744c52e4221163991fac90275354db1c795","schema_version":"1.0","event_id":"sha256:dae5c3d00db4d39984bec0ff8f078744c52e4221163991fac90275354db1c795"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NCWYHB5KPIKQAS3G4WFKKFIJSM/bundle.json","state_url":"https://pith.science/pith/NCWYHB5KPIKQAS3G4WFKKFIJSM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NCWYHB5KPIKQAS3G4WFKKFIJSM/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":"2026-06-05T06:10:54Z","links":{"resolver":"https://pith.science/pith/NCWYHB5KPIKQAS3G4WFKKFIJSM","bundle":"https://pith.science/pith/NCWYHB5KPIKQAS3G4WFKKFIJSM/bundle.json","state":"https://pith.science/pith/NCWYHB5KPIKQAS3G4WFKKFIJSM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NCWYHB5KPIKQAS3G4WFKKFIJSM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NCWYHB5KPIKQAS3G4WFKKFIJSM","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":"be8371a20069eacc8f71e5ccebd9f06d9f699e6411dff3dc41f66f2133e77dcf","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-18T16:51:26Z","title_canon_sha256":"820725a3250158ed1828cd4aaba096d5196ca7138837ed4e792031e1c96802bd"},"schema_version":"1.0","source":{"id":"2605.18644","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.18644","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"arxiv_version","alias_value":"2605.18644v1","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.18644","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"pith_short_12","alias_value":"NCWYHB5KPIKQ","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"pith_short_16","alias_value":"NCWYHB5KPIKQAS3G","created_at":"2026-05-20T00:06:12Z"},{"alias_kind":"pith_short_8","alias_value":"NCWYHB5K","created_at":"2026-05-20T00:06:12Z"}],"graph_snapshots":[{"event_id":"sha256:dae5c3d00db4d39984bec0ff8f078744c52e4221163991fac90275354db1c795","target":"graph","created_at":"2026-05-20T00:06:12Z","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":[{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-20T00:01:59.181012Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.18644/integrity.json","findings":[],"snapshot_sha256":"ac61f50557c1259a0687e5bb151feb71301f66915f4bc11434dbe7e70bea9d81","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \\geq b \\geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete description of all such quadruples when $m=2,3,4,5$, or $6$, except when $m=6$ and $b=c=1$. We also show that if the LCM of the moduli has only $2$, $3$, or $5$ as prime divisors, then $m \\leq 9$ and construct a distinct covering system with $m=8$, $a=8$, $b=3$, and $c=2$. When a covering system exists for a quadruple $(m,a,b,c)$ we provide an example. Nonexisten","authors_text":"Jonah Klein, Joshua Harrington, Joshua Lowrance, Ognian Trifonov","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-18T16:51:26Z","title":"Covering systems where the prime divisors of all moduli are only $2$, $3$, or $5$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.18644","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:b674fb1f47972d529feab7065e5659d6961b82e65db80e9df33b4f31cc0692b6","target":"record","created_at":"2026-05-20T00:06:12Z","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":"be8371a20069eacc8f71e5ccebd9f06d9f699e6411dff3dc41f66f2133e77dcf","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-18T16:51:26Z","title_canon_sha256":"820725a3250158ed1828cd4aaba096d5196ca7138837ed4e792031e1c96802bd"},"schema_version":"1.0","source":{"id":"2605.18644","kind":"arxiv","version":1}},"canonical_sha256":"68ad8387aa7a15004b66e58aa51509933175b3369348f9003f90919e4d4fcd94","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68ad8387aa7a15004b66e58aa51509933175b3369348f9003f90919e4d4fcd94","first_computed_at":"2026-05-20T00:06:12.536703Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:06:12.536703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Yo9gLKBs1RcCOofJawjctbuvDQmZgw8AAEz5McGmQswexG5XPoy5cOPVBoI40iSwLrzn2ZQ69OcnaFdx/NN+AQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:06:12.537502Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.18644","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b674fb1f47972d529feab7065e5659d6961b82e65db80e9df33b4f31cc0692b6","sha256:dae5c3d00db4d39984bec0ff8f078744c52e4221163991fac90275354db1c795"],"state_sha256":"9aab17c588d5d4532b9d508275711f181d58295a76fd4ba33b6e70f8dbf7e774"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ccx2sXauwzAScDUezDV7+XvU+PU5o7HRy4GDBzDFRrsl2zkXyzdj1S1HTPy4uPJwnsHpaAjxDl40OBxb6fUoAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T06:10:54.716945Z","bundle_sha256":"bf7b41fb2317eba26ce88d7f28bb6314843d8f32d6c5c4e272fd579fd5d584dc"}}