Erdős problem 851 · problems.science

Let $\epsilon > 0$ . Is there some $r \ll_\epsilon 1$ such that the density of integers of the form $2^k+n$ , where $k \geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\epsilon$ ?

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FormalConjectures/ErdosProblems/

851.lean

Retained formal statement1 of 2

Let $\epsilon > 0$ . Is there some $r \ll_\epsilon 1$ such that the density of integers of the form $2^k+n$ , where $k \geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\epsilon$ ?

This was proved affirmatively by Price and GPT-5.2 Pro [Pr26].

FormalConjectures/ErdosProblems/851.leanErdos851.erdos_8511 lineExact file

1∀ ε ∈ Set.Ioo 0 1, ∃ r d, (Erdos851.TwoPowAddSet r).HasDensity d ∧ 1 - ε ≤ d

SolvedStatement only, no proof

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