{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:AADT3F7E7JQKPZGENMLMKQUAQV","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":"b7a0e4e93a5e524330fe5c57699e5e25a11d2d4effd27a5632659fa5800f6d0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-30T17:49:15Z","title_canon_sha256":"bfbfdfaf2942e1fcfa7cce29ce06bc9ba6724daaab1795c88d64f468d6af04a5"},"schema_version":"1.0","source":{"id":"1309.7941","kind":"arxiv","version":7}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7941","created_at":"2026-05-18T01:48:06Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7941v7","created_at":"2026-05-18T01:48:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7941","created_at":"2026-05-18T01:48:06Z"},{"alias_kind":"pith_short_12","alias_value":"AADT3F7E7JQK","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AADT3F7E7JQKPZGE","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AADT3F7E","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:f158122c786aefba2c5b326e858267c2b7ff5f48783c316ec28df8acc650b3a8","target":"graph","created_at":"2026-05-18T01:48:06Z","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":"We show that if the congruence above holds and $n\\mid m$, then the quotient $Q:=m/n$ satisfies $\\sum_{p\\mid Q} \\frac{Q}{p}+1 \\equiv 0\\pmod{Q}$, where $p$ is prime. The only known solutions of the latter congruence are $Q=1$ and the eight known primary pseudoperfect numbers $2,6,42, 1806, 47058, 2214502422, 52495396602,$ and $8490421583559688410706771261086$. Fixing $Q$, we prove that the set of positive integers $n$ satisfying the congruence in the title, with $m=Q n$, is empty in case $Q=52495396602$, and in the other eight cases has an asymptotic density between bounds in $(0,1)$ that we pro","authors_text":"Antonio M. Oller-Marc\\'en, Jonathan Sondow, Jos\\'e Mar\\'ia Grau","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-30T17:49:15Z","title":"On the congruence $1^m + 2^m + \\dotsb + m^m \\equiv n \\pmod{m}$ with $n | m$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7941","kind":"arxiv","version":7},"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:cc46ef01fbe4ec2cf24f63ecbafaf08fd532a9671e99377f9a514875441a24e0","target":"record","created_at":"2026-05-18T01:48:06Z","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":"b7a0e4e93a5e524330fe5c57699e5e25a11d2d4effd27a5632659fa5800f6d0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-30T17:49:15Z","title_canon_sha256":"bfbfdfaf2942e1fcfa7cce29ce06bc9ba6724daaab1795c88d64f468d6af04a5"},"schema_version":"1.0","source":{"id":"1309.7941","kind":"arxiv","version":7}},"canonical_sha256":"00073d97e4fa60a7e4c46b16c54280856b3e5b89955b3443426d117dc0f3dfae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"00073d97e4fa60a7e4c46b16c54280856b3e5b89955b3443426d117dc0f3dfae","first_computed_at":"2026-05-18T01:48:06.909794Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:48:06.909794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wEmAJQikaT52bT2hVtp3MvvHeI9WLJ4LBYPu0hf2illsbNZjc7gytshHyi5jX/JRYe5Yflfpc3+s+z/PVoYUAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:48:06.910301Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.7941","source_kind":"arxiv","source_version":7}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cc46ef01fbe4ec2cf24f63ecbafaf08fd532a9671e99377f9a514875441a24e0","sha256:f158122c786aefba2c5b326e858267c2b7ff5f48783c316ec28df8acc650b3a8"],"state_sha256":"7234884be58d3ff86155575cf589d91b47ab38dd1135a780fcdb4460dc14b940"}