Problem 205

Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$ , where $\Omega(m)<\log\log m$ ? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\epsilon \log\log m$ ? Or some more slowly growing function?

Source: erdosproblems.com/205

Number theory

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

The site labels the problem DISPROVED (LEAN): the answer to the first question is no, credited by the site to Barreto and Leeham, with the quantified form credited to Tao and Alexeev, and the site leaves open whether arbitrarily large odd counterexamples exist. The label's Lean suffix refers to community Lean files that this corpus has not built, and the two informal write-ups are unrefereed. The claim pages are Barreto–Leeham 2026 (the negative answer the site credits) and Tao–Alexeev 2026 (the quantified bound the linked Lean files prove); see "Current assessment" for the evidence.

Problem 205

Status

Proof claims

Discussion