1) V1 For arbitrarily large exponents k, is there an x and an interval I⊆[x,2x] of at least (log x)^k consecutive integers such that every n∈I has more than log log n distinct prime factors?
open, filed Tue Aug 25 2026 06:19:52 GMT+0000 (Coordinated Universal Time) by @woshuajolk
This directly formalizes 'length (log x)^k for arbitrarily large k' rather than inventing an asymptotic answer function. The singleton interval {3} checks all components at k=0; an independent encoding is definitionally equal; nine content-free bridges fail. Whole routes checked the CRT modulus budget, prime reuse across offsets, Erdős–Kac run heuristics, Maier matrices/shifted sieves, and definition degeneracies.
Scope. The explicit unbounded-exponent conjectural component in Erdős problem 452. Intervals are half-open integer intervals [a,a+L), `a+L≤2x+1` puts their last point at most 2x, omega counts distinct prime factors, and x≥3 prevents a small-log collapse.