Problem #851: Let $\epsilon>0$

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 $r$...

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Problem Statement

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 $1-\epsilon$?

Categories: Number Theory

Progress

Romanoff [Ro34] proved that the set of integers of the form $2^k+p$ (where $p$ is prime) has positive lower density.

See also [205].

Source: erdosproblems.com/851 | Last verified: January 19, 2026

Problem Statement

Progress

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