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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 ","authors_text":"Christian Elsholtz, Li-Li Jiang, Yong-Gao Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-31T03:08:56Z","title":"Egyptian Fractions with Restrictions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.6118","kind":"arxiv","version":1},"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":"sha256:8eae5cba1dc87c0c1c90df71ad7ba665baabe9bbeb983005e93fd2e86d6c8b99","target":"record","created_at":"2026-05-18T02:42:54Z","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":"ea9941a0ae64686d381c540365a51835768f65135ddc37332fec9e6bbefba4b4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-31T03:08:56Z","title_canon_sha256":"21735d599f4be1c5fae3f85a7d3cdf6fc160796fd4050b179601878edadfa90d"},"schema_version":"1.0","source":{"id":"1108.6118","kind":"arxiv","version":1}},"canonical_sha256":"cb03c2ddb575db29a6c6507bf01d1fa052c0b6139ab3cf35d6dc73abcd75efe0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb03c2ddb575db29a6c6507bf01d1fa052c0b6139ab3cf35d6dc73abcd75efe0","first_computed_at":"2026-05-18T02:42:54.389988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:54.389988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SDUYcOYxlewKzm7tTmp3nFoNlbzDEmr0AeLy87SPG/xULg7rTg7ocsIucGPZ1vQz4Z4lwU3/dLu/Z+C3MimvDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:54.390382Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.6118","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8eae5cba1dc87c0c1c90df71ad7ba665baabe9bbeb983005e93fd2e86d6c8b99","sha256:77c60ceb120b3f6c1ee77fbb7e5ea1d3f40c0710a309c4cb8202b1995b45a671"],"state_sha256":"a0a4ef12fa6b8abab3054999d62f93d94c4e6c891f9994faf16be4a308543729"}