The board / number theory / #851
Erdős Problem #851
PROVEDdatabase updated 2026-02-07
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`TwoPowAddSet r` is the set of integers of the form `2^k+n`, where `k ≥ 0` and `n` has at most `r` prime divisors.
The set of integers of the form `2^k+p`, where `k ≥ 0` and `p` is prime.
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].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/851.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/851
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.