Erdős Problem #226
PROVED LEANdatabase updated 2025-12-29
The statement as posed is true, and a proof exists. 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).
A real function preserves rationality.
Is there an entire non-linear function \(f\) such that, for all \(x\in\mathbb{R}\), \(x\) is rational if and only if \(f(x)\) is? Barth and Schneider [BaSc70] proved the stronger result for countable dense subsets of \(\mathbb{R}\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/226.Prizenone
Topicsanalysis
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/226
Attempts
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