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We will develop some techniques that allow one to exclude many integers $\\rho$ as a ratio and combine them to exclude the integers $3\\le \\rho\\le 1501$ and, assuming a conjecture on irregular primes to be true, a set of density $1$ of ratios $\\rho$. To exclude a ratio $\\rho$ one has to show that the Erd\\H{o}s-Moser type equation $(\\rho-1)S_k(m)=m^k$ has no non-trivial solutions."},"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":"1510.06064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-20T21:16:20Z","cross_cats_sorted":[],"title_canon_sha256":"225fd39e09010ea185f258b7fd082b0efb95e939239e2d1f32e417e040812255","abstract_canon_sha256":"864a5ec52b7ebb4e5e9bf4c7591b58723a9fdbd8e00673a58b86adff004cec7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:12.523481Z","signature_b64":"senuT4PsHlxCrrUZJRgxfeHM0crOMZYsRoBaRR0xRCKA2cO9gfy0LE1Y4N42eroh4o3VGoV+bwdEZLr60sgkBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d38a0981e49753b12774d3d3f6dce52ec39a88e1c8b5b3185b5a96166b91bef2","last_reissued_at":"2026-05-18T00:53:12.522933Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:12.522933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Forbidden integer ratios of consecutive power sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ioulia N. 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