The board / number theory / #999
Erdős Problem #999
Duffin-Schaeffer conjecture
PROVEDdatabase updated 2025-09-07
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).
`x` is *approximable* with respect to `f` if \[\left\lvert x-\frac{p}{q}\right\rvert < \frac{f(q)}{q}\] has infinitely many solutions with \((p,q)=1\).
For any function \(f:\mathbb{N}\to \mathbb{R}_{\geq 0}\) the property that, for almost all \(\alpha\) \[\left\lvert \alpha-\frac{p}{q}\right\rvert < \frac{f(q)}{q}\] has infinitely many solutions with \((p,q)=1\), is equivalent to \[\sum_{q\geq 1}\phi(q)\frac{f(q)}{q}=\infty.\] The Duffin–Schaeffer conjecture. It is easy to prove that the latter follows from the former. Erdős proved this in the special case when \(f(q)q\) is bounded. The full conjecture was proved by Koukoulopoulos and Maynard [KoMa20].
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/999.Attempts
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