Liouville Classification of the Gelfond Constant

Definitions and mathematical statement

A real number $x$ is a Liouville number if it is irrational and \[\forall m\in{\mathbb{Z}}_{\geq1}\ \exists p\in{\mathbb{Z}}\ \exists q\in{\mathbb{Z}}_{\geq2}:\qquad 0<\left|x-\frac pq\right|<q^{-m}.\] Let $e$ be the base of the natural logarithm and let $\pi$ be the usual circle constant. The source asks whether $e^\pi$ is a Liouville number and explicitly conjectures the negative answer: \[e^\pi\text{ is not a Liouville number}.\] In view of the transcendence reported in the supplied literature assessment, the conjecture may equivalently be written \[\exists m\in{\mathbb{Z}}_{\geq1}\ \forall p\in{\mathbb{Z}}\ \forall q\in{\mathbb{Z}}_{\geq2}:\qquad \left|e^\pi-\frac pq\right|\geq q^{-m}.\] The source's literature triage, dated 2026-08-21, labels the classification open and reports known transcendence and quantitative approximation bounds. It does not state those bounds, and they are not treated here as establishing the conjectured non-Liouville property.

Short English statement

Determine whether $e^\pi$ admits rational approximations of arbitrarily large power-law accuracy. The report conjectures that it does not, and the supplied 2026-08-21 assessment labels this open.

Sources

• Local UnsolvedMath registry, permanent ID 30000754, source problem catalogue entry, Liouville Classification of the Gelfond Constant, proposed year 2007; no individual proposer supplied.
• Supplied report citation: Diophantische Approximationen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/21, pp. 1115--1190, https://doi.org/10.4171/owr/2007/21.
• Supplied literature label: Oberwolfach report passage, https://ems.press/content/serial-article-files/46108.
• Supplied literature label: Waldschmidt, Schanuel conjecture notes, https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf.