{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:LB6VXT4W4Q6LSA7Y6WACGFWUKA","short_pith_number":"pith:LB6VXT4W","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"},"canonical_sha256":"587d5bcf96e43cb903f8f5802316d45021244c82679dcaef22ab0e7a1c26dbed","source":{"kind":"arxiv","id":"1311.1389","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.1389","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"arxiv_version","alias_value":"1311.1389v2","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.1389","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"pith_short_12","alias_value":"LB6VXT4W4Q6L","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LB6VXT4W4Q6LSA7Y","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LB6VXT4W","created_at":"2026-05-18T12:27:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:LB6VXT4W4Q6LSA7Y6WACGFWUKA","target":"record","payload":{"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"},"canonical_sha256":"587d5bcf96e43cb903f8f5802316d45021244c82679dcaef22ab0e7a1c26dbed","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"},"source_kind":"arxiv","source_id":"1311.1389","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8LWF2huAIKJceZjIojzhCRjdCyYO5BSuk+Dt1+2xOM4SpY1q8JHrwBMe8o+6L15ny0tqLOZ9tn/6htuqDHJwBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:08:36.918923Z"},"content_sha256":"b44cac76eaafc6c520655376dc0c3e8a6cf881b1609dffc0f6d1b2d60703e401","schema_version":"1.0","event_id":"sha256:b44cac76eaafc6c520655376dc0c3e8a6cf881b1609dffc0f6d1b2d60703e401"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:LB6VXT4W4Q6LSA7Y6WACGFWUKA","target":"graph","payload":{"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. In 1946, Erd\\H{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1389","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oXRChQgw2dXKSrCQwIHiT8zwXmyFcfqaoJHuIAaW7OFIqhjG7RhB96g0YHdaQBV3PconNgkUF5edZuMXqWyjCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:08:36.919262Z"},"content_sha256":"979dd9d9ebb26520f91e3a3f99e81b10c2c82c9cd644fbebcf41f2a497f129a8","schema_version":"1.0","event_id":"sha256:979dd9d9ebb26520f91e3a3f99e81b10c2c82c9cd644fbebcf41f2a497f129a8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LB6VXT4W4Q6LSA7Y6WACGFWUKA/bundle.json","state_url":"https://pith.science/pith/LB6VXT4W4Q6LSA7Y6WACGFWUKA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LB6VXT4W4Q6LSA7Y6WACGFWUKA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T07:08:36Z","links":{"resolver":"https://pith.science/pith/LB6VXT4W4Q6LSA7Y6WACGFWUKA","bundle":"https://pith.science/pith/LB6VXT4W4Q6LSA7Y6WACGFWUKA/bundle.json","state":"https://pith.science/pith/LB6VXT4W4Q6LSA7Y6WACGFWUKA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LB6VXT4W4Q6LSA7Y6WACGFWUKA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:LB6VXT4W4Q6LSA7Y6WACGFWUKA","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":"fa916e3f3b87d756dfc8ca0992b93cade0f402fbdab3355b6a1a71ff05c820cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-06T13:43:46Z","title_canon_sha256":"b9575e3936f2a3ba6781420a5959d988825a7306a9f7257b1903849bc2988a47"},"schema_version":"1.0","source":{"id":"1311.1389","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.1389","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"arxiv_version","alias_value":"1311.1389v2","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.1389","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"pith_short_12","alias_value":"LB6VXT4W4Q6L","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LB6VXT4W4Q6LSA7Y","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LB6VXT4W","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:979dd9d9ebb26520f91e3a3f99e81b10c2c82c9cd644fbebcf41f2a497f129a8","target":"graph","created_at":"2026-05-18T02:55:48Z","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":"Let $a$ and $b$ be positive integers. In 1946, Erd\\H{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. 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.","authors_text":"Chunlin Wang, Shaofang Hong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-06T13:43:46Z","title":"The elementary symmetric functions of reciprocals of the elements of arithmetic progressions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1389","kind":"arxiv","version":2},"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:b44cac76eaafc6c520655376dc0c3e8a6cf881b1609dffc0f6d1b2d60703e401","target":"record","created_at":"2026-05-18T02:55:48Z","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":"fa916e3f3b87d756dfc8ca0992b93cade0f402fbdab3355b6a1a71ff05c820cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-06T13:43:46Z","title_canon_sha256":"b9575e3936f2a3ba6781420a5959d988825a7306a9f7257b1903849bc2988a47"},"schema_version":"1.0","source":{"id":"1311.1389","kind":"arxiv","version":2}},"canonical_sha256":"587d5bcf96e43cb903f8f5802316d45021244c82679dcaef22ab0e7a1c26dbed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"587d5bcf96e43cb903f8f5802316d45021244c82679dcaef22ab0e7a1c26dbed","first_computed_at":"2026-05-18T02:55:48.226882Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:48.226882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s7a5IvYNhCIYg72YDNGYQypazIXRm6Xej3IyH8lLSJGsIkTE2n8mNE0iNoQV5mpG7KRMB/FG9aGpjs4hUF+PAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:48.227655Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.1389","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b44cac76eaafc6c520655376dc0c3e8a6cf881b1609dffc0f6d1b2d60703e401","sha256:979dd9d9ebb26520f91e3a3f99e81b10c2c82c9cd644fbebcf41f2a497f129a8"],"state_sha256":"e35eb22901889ebfec50fde0122f3b013b9427aed4c84c97835c7dcab02f9321"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"84atIoILpshD8e/VycbHgPGua4NLs5ZgMdHoOWot8FukOhtO1BVZIAb2YpE1dV50zumRkItCMYgxyxvHGd/YCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T07:08:36.921208Z","bundle_sha256":"3a17fc9aacef3397ac453a86f2a0384efdc400bac5258fa64917b512c96264e1"}}