The board / number theory / #205
Erdős Problem #205
DISPROVED LEANdatabase updated 2026-01-10
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`IsRepresentable f n` states that `n` can be written as `2 ^ k + m` for some `k ≥ 0` where the number of prime divisors of `m`, counted with multiplicity, is less than `f m`.
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.) Barreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was quantified by Tao and Alexeev (see the comments): in fact there are infinitely many \(n\) such that, for all \(k\) with \(2^k<n\), \(n-2^k\) has at least \[\gg \left(\frac{\log n}{\log\log n}\right)^{1/2}\] many prime factors.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/205.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/205
Attempts
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