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In this paper, we show that for many integers $a$ the equation $aT_k(m)=(2m+1)^k$, where $T_k(m):=\\sum_{j=1}^m(2j-1)^k$, has no solutions in positive integers $k$ and $m$. This leads us to the conjecture that for $m>1$ the ratio $T_k(m+1)/T_k(m)$ is never an integer."},"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":"1804.04646","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-12T17:46:43Z","cross_cats_sorted":[],"title_canon_sha256":"5c1e2977fc08de5b0b3742d1442d0c489652dc995bd75fe7c13aae52424effa0","abstract_canon_sha256":"1e2e13e13bd71623c8c6977f5ca71b36fc9c9565096c01301b9f786a51e259b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:41.393946Z","signature_b64":"FgCoel9Jr+aEEStfZOAh0xxMS0PQ6mTxrDC76rEo5xq+97lCDszhYF1fBF7Wtz84kGQdS2rH+Abcvv98kNCqBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"920fabca972ae572db39b87ae8ff70562d2f0fef24ad35cccc47f022bb38cfec","last_reissued_at":"2026-05-17T23:56:41.393579Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:41.393579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the unsolvability of certain equations of Erd\\H{o}s-Moser type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ioulia N. Baoulina","submitted_at":"2018-04-12T17:46:43Z","abstract_excerpt":"Let $S_k(m):=\\sum_{j=1}^{m-1}j^k$ denote a power sum. In 2011, Kellner proposed the conjecture that for $m>3$ the ratio $S_k(m+1)/S_k(m)$ is never an integer, or, equivalently, that for any positive integer $a$, the equation $aS_k(m)=m^k$ has no solutions in positive integers $k$ and $m$ with $m>3$. In this paper, we show that for many integers $a$ the equation $aT_k(m)=(2m+1)^k$, where $T_k(m):=\\sum_{j=1}^m(2j-1)^k$, has no solutions in positive integers $k$ and $m$. 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