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In this paper, we show that for any integer $k$ with $1\\le k\\le n$, the $k$-th elementary symmetric function of $1/b, 1/(a+b),..., 1/(an-a+b)$ is not an integer except that either $b=n=k=1$ and $a\\ge 1$, or $a=b=1, n=3$ and $k=2$. This refines the Erd\\H{o}s-Niven theorem and answers an open problem raised by Chen and Tang in 2012."},"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":"1311.1389","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-06T13:43:46Z","cross_cats_sorted":[],"title_canon_sha256":"b9575e3936f2a3ba6781420a5959d988825a7306a9f7257b1903849bc2988a47","abstract_canon_sha256":"fa916e3f3b87d756dfc8ca0992b93cade0f402fbdab3355b6a1a71ff05c820cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:48.227655Z","signature_b64":"s7a5IvYNhCIYg72YDNGYQypazIXRm6Xej3IyH8lLSJGsIkTE2n8mNE0iNoQV5mpG7KRMB/FG9aGpjs4hUF+PAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"587d5bcf96e43cb903f8f5802316d45021244c82679dcaef22ab0e7a1c26dbed","last_reissued_at":"2026-05-18T02:55:48.226882Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:48.226882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The elementary symmetric functions of reciprocals of the elements of arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chunlin Wang, Shaofang Hong","submitted_at":"2013-11-06T13:43:46Z","abstract_excerpt":"Let $a$ and $b$ be positive integers. 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