{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:ZMB4FXNVOXNSTJWGKB57AHI7UB","short_pith_number":"pith:ZMB4FXNV","schema_version":"1.0","canonical_sha256":"cb03c2ddb575db29a6c6507bf01d1fa052c0b6139ab3cf35d6dc73abcd75efe0","source":{"kind":"arxiv","id":"1108.6118","version":1},"attestation_state":"computed","paper":{"title":"Egyptian Fractions with Restrictions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian Elsholtz, Li-Li Jiang, Yong-Gao Chen","submitted_at":"2011-08-31T03:08:56Z","abstract_excerpt":"Let $T_o(k)$ denote the number of solutions of $\\sum_{i=1}^k\\frac 1{x_i}=1$ in odd numbers $1<x_1<x_2<...<x_k$. It is clear that $T_o(2k)=0$. For distinct primes $p_1, p_2,..., p_t$, let $S(p_1, p_2,..., p_t)=\\{p_1^{\\alpha_1}...p_t^{\\alpha_t}\\mid \\alpha_i\\in \\mathbb{N}_0, i=1,2,..., t}$. Let $T_k(p_1,..., p_t)$ be the number of solutions $\\sum_{i=1}^{k}\\frac 1{x_i}=1$ with $1<x_1<x_2<...<x_{k}$ and $x_i\\in S(p_1, p_2,..., p_t)$. It is clear that if $T_k(p_1,..., p_t)\\not= 0$ for some $k$, then the inverse sum of all elements $s_j>1$ in $S(p_1, p_2,..., p_t)$ is more than 1.\n  In this paper we "},"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":"1108.6118","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-31T03:08:56Z","cross_cats_sorted":[],"title_canon_sha256":"21735d599f4be1c5fae3f85a7d3cdf6fc160796fd4050b179601878edadfa90d","abstract_canon_sha256":"ea9941a0ae64686d381c540365a51835768f65135ddc37332fec9e6bbefba4b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:54.390382Z","signature_b64":"SDUYcOYxlewKzm7tTmp3nFoNlbzDEmr0AeLy87SPG/xULg7rTg7ocsIucGPZ1vQz4Z4lwU3/dLu/Z+C3MimvDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb03c2ddb575db29a6c6507bf01d1fa052c0b6139ab3cf35d6dc73abcd75efe0","last_reissued_at":"2026-05-18T02:42:54.389988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:54.389988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Egyptian Fractions with Restrictions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian Elsholtz, Li-Li Jiang, Yong-Gao Chen","submitted_at":"2011-08-31T03:08:56Z","abstract_excerpt":"Let $T_o(k)$ denote the number of solutions of $\\sum_{i=1}^k\\frac 1{x_i}=1$ in odd numbers $1<x_1<x_2<...<x_k$. It is clear that $T_o(2k)=0$. For distinct primes $p_1, p_2,..., p_t$, let $S(p_1, p_2,..., p_t)=\\{p_1^{\\alpha_1}...p_t^{\\alpha_t}\\mid \\alpha_i\\in \\mathbb{N}_0, i=1,2,..., t}$. Let $T_k(p_1,..., p_t)$ be the number of solutions $\\sum_{i=1}^{k}\\frac 1{x_i}=1$ with $1<x_1<x_2<...<x_{k}$ and $x_i\\in S(p_1, p_2,..., p_t)$. It is clear that if $T_k(p_1,..., p_t)\\not= 0$ for some $k$, then the inverse sum of all elements $s_j>1$ in $S(p_1, p_2,..., p_t)$ is more than 1.\n  In this paper we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.6118","kind":"arxiv","version":1},"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":"1108.6118","created_at":"2026-05-18T02:42:54.390058+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.6118v1","created_at":"2026-05-18T02:42:54.390058+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.6118","created_at":"2026-05-18T02:42:54.390058+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZMB4FXNVOXNS","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZMB4FXNVOXNSTJWG","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZMB4FXNV","created_at":"2026-05-18T12:26:47.523578+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB","json":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB.json","graph_json":"https://pith.science/api/pith-number/ZMB4FXNVOXNSTJWGKB57AHI7UB/graph.json","events_json":"https://pith.science/api/pith-number/ZMB4FXNVOXNSTJWGKB57AHI7UB/events.json","paper":"https://pith.science/paper/ZMB4FXNV"},"agent_actions":{"view_html":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB","download_json":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB.json","view_paper":"https://pith.science/paper/ZMB4FXNV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.6118&json=true","fetch_graph":"https://pith.science/api/pith-number/ZMB4FXNVOXNSTJWGKB57AHI7UB/graph.json","fetch_events":"https://pith.science/api/pith-number/ZMB4FXNVOXNSTJWGKB57AHI7UB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB/action/storage_attestation","attest_author":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB/action/author_attestation","sign_citation":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB/action/citation_signature","submit_replication":"https://pith.science/pith/ZMB4FXNVOXNSTJWGKB57AHI7UB/action/replication_record"}},"created_at":"2026-05-18T02:42:54.390058+00:00","updated_at":"2026-05-18T02:42:54.390058+00:00"}