Problem 226

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claims/: The 1 claim page of Problem 226, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Status. PROVED (LEAN). The site credits Barth and Schneider [BaSc70], whose theorem for arbitrary countable dense subsets of the reals gives the function at $A=B=\mathbb Q$; the claim page [[problems/analysis/E0226/claims/1970_10_01_barth_schneider|Barth and Schneider 1970]] records it, accepted on its refereed publication and the site's credit. The site's Lean qualification refers to Boris Alexeev's formalization of that solution, linked from the claim page and not built by this corpus.

Source. erdosproblems.com/226, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #226, https://www.erdosproblems.com/226.

References.

• [BaSc70] Barth, K. F. and Schneider, W. J., Entire functions mapping countable dense subsets of the reals onto each other monotonically. J. London Math. Soc. (2) (1970), 620-626.
• [BaSc71] Barth, K. F. and Schneider, W. J., Entire functions mapping arbitrary countable dense sets and their complements onto each other. J. London Math. Soc. (2) (1971/72), 482-488.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.

Formalization. Statement in formal-conjectures, whose formal_proof attribute points at a Lean 4 proof in Boris Alexeev's lean-proofs repository, recorded on the claim page as a formalization of Barth and Schneider's solution; this project has not built it.

Current assessment

The site's formulation of 2026-09-04 asks for an entire function, not linear, that is rational at exactly the rational reals. Barth and Schneider's 1970 theorem produces a transcendental entire function, real on the real line, that maps a prescribed countable dense set $A\subset\mathbb R$ onto a prescribed countable dense set $B$ and no other real point into $B$; with $A=B=\mathbb Q$ it answers the question yes, and their 1972 sequel does the same for countable dense subsets of $\mathbb C$. The standing rests on the refereed paper and the site's credit, as the claim page records. The paper is not held in the library, so this corpus compiles and reviews no proof, and the Lean development the site's label refers to has not been built or audited by this project. No status search beyond the site is recorded.