{"id":"palomar/PALOMAR-2026-10-06-000001","url":"https://github.com/linrock/math-proofs/tree/38ad7ea8de1efe526f665c2b5204465e5ed879fe/erdos-1139","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-06-000001-v1.json","repo":"https://github.com/linrock/math-proofs","tags":["math.NT","math.CO","msc:11N05","msc:11N36","msc:11B25","analytic-number-theory","almost-primes","erdos-problems","lean-formalization"],"score":0,"title":"Erdős 1139: large gaps between almost-primes","trust":"high","domain":"math.NT","source":"palomar","authors":["Linmiao Xu"],"reviews":0,"subject":"math.NT","summary":"A Lean 4 formalization of unconditional bounds and conditional reductions for Erdős problem 1139 on consecutive P_2 almost-primes (Ω(u_k) ≤ 2). Extending Chojecki's (2026) Erdős 689 double-covering construction via protected-reserve omission, it proves unconditionally that limsup (u_{k+1} - u_k) / log(k + 1) ≥ A > 1 and that every large height X contains an almost-prime-free interval of length ≥ c log X (c > 1), alongside the mixed prime/prime-square CRT bridge, PNT primorial asymptotics, and squared-core deficiency classification. Using a squared-prime core and scale-adaptive dyadic prime…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-06","reactions":{},"formalized":true,"formalizes":[{"title":"A greedy matching proof of Erdős's two-fold residue-class problem","identifier":"https://www.ulam.ai/research/erdos689.pdf","relationship":"adapts"}]}