{"paper":{"title":"Unit fractions with semiprime denominators: an elementary proof of Erd\\H{o}s Problem #306","arxiv_id":"2609.32140","primary_cat":"math.NT","submitted_at":"2026-09-26T01:53:41+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.55,"id":"s0","own":true,"role":"contribution","text":"We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erd\\H{o}s Problem #306).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.97,"quantitative":false,"quantitative_p":0.25,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.45,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.55,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"method","text":"After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":2,"p":0.99,"id":"s2","own":true,"role":"method","text":"Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph.","hedge":{"p":0.95,"label":"asserted"},"own_p":0.92,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":3,"p":0.98,"id":"s3","own":true,"role":"method","text":"The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.71,"role_probabilities":{"gap":0.0,"method":0.98,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":4,"p":0.68,"id":"s4","own":true,"role":"method","text":"The only inputs about primes are Chebyshev-type bounds.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.53,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.29,"role_probabilities":{"gap":0.0,"method":0.68,"result":0.01,"background":0.02,"limitation":0.16,"future_work":0.0,"contribution":0.09,"interpretation":0.04}},{"i":5,"p":0.62,"id":"s5","own":true,"role":"background","text":"The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.61,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.28,"role_probabilities":{"gap":0.0,"method":0.11,"result":0.0,"background":0.62,"limitation":0.0,"future_work":0.0,"contribution":0.27,"interpretation":0.0}},{"i":6,"p":0.7,"id":"s6","own":true,"role":"limitation","text":"The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.82,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.25,"role_probabilities":{"gap":0.0,"method":0.07,"result":0.01,"background":0.0,"limitation":0.7,"future_work":0.0,"contribution":0.22,"interpretation":0.0}},{"i":7,"p":0.47,"id":"s7","own":true,"role":"method","text":"This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.6,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.47,"result":0.0,"background":0.28,"limitation":0.03,"future_work":0.0,"contribution":0.21,"interpretation":0.01}}]},"compiled":{"main_claim":{"p":1.0,"text":"We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erd\\H{o}s Problem #306).","location":"abstract s0","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":6423,"cost_usd_reported":0.0},"compiled_at":"2026-09-29","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.1,"states_limitation":0.9,"human_participants":0.21,"controlled_comparison":0.02,"randomized_assignment":0.02,"states_quantitative_result":0.19,"compares_with_prior_results":0.82,"mentions_code_or_data_availability":0.16},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.58,"text":"We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erd\\H{o}s Problem #306).","location":"abstract s0"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":8,"sentences_in_abstract":8},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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