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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"},"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":"1309.7941","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-30T17:49:15Z","cross_cats_sorted":[],"title_canon_sha256":"bfbfdfaf2942e1fcfa7cce29ce06bc9ba6724daaab1795c88d64f468d6af04a5","abstract_canon_sha256":"b7a0e4e93a5e524330fe5c57699e5e25a11d2d4effd27a5632659fa5800f6d0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:48:06.910301Z","signature_b64":"wEmAJQikaT52bT2hVtp3MvvHeI9WLJ4LBYPu0hf2illsbNZjc7gytshHyi5jX/JRYe5Yflfpc3+s+z/PVoYUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00073d97e4fa60a7e4c46b16c54280856b3e5b89955b3443426d117dc0f3dfae","last_reissued_at":"2026-05-18T01:48:06.909794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:48:06.909794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the congruence $1^m + 2^m + \\dotsb + m^m \\equiv n \\pmod{m}$ with $n | m$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonio M. 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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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7941","kind":"arxiv","version":7},"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":"1309.7941","created_at":"2026-05-18T01:48:06.909875+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.7941v7","created_at":"2026-05-18T01:48:06.909875+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7941","created_at":"2026-05-18T01:48:06.909875+00:00"},{"alias_kind":"pith_short_12","alias_value":"AADT3F7E7JQK","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"AADT3F7E7JQKPZGE","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"AADT3F7E","created_at":"2026-05-18T12:27:38.830355+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.21518","citing_title":"Port Fillings for Primary Pseudoperfect Numbers","ref_index":10,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV","json":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV.json","graph_json":"https://pith.science/api/pith-number/AADT3F7E7JQKPZGENMLMKQUAQV/graph.json","events_json":"https://pith.science/api/pith-number/AADT3F7E7JQKPZGENMLMKQUAQV/events.json","paper":"https://pith.science/paper/AADT3F7E"},"agent_actions":{"view_html":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV","download_json":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV.json","view_paper":"https://pith.science/paper/AADT3F7E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.7941&json=true","fetch_graph":"https://pith.science/api/pith-number/AADT3F7E7JQKPZGENMLMKQUAQV/graph.json","fetch_events":"https://pith.science/api/pith-number/AADT3F7E7JQKPZGENMLMKQUAQV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV/action/storage_attestation","attest_author":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV/action/author_attestation","sign_citation":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV/action/citation_signature","submit_replication":"https://pith.science/pith/AADT3F7E7JQKPZGENMLMKQUAQV/action/replication_record"}},"created_at":"2026-05-18T01:48:06.909875+00:00","updated_at":"2026-05-18T01:48:06.909875+00:00"}