Conjecture 2.15 — Mahler

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(\varepsilon_n)_{n\ge0}$ be any sequence with $\varepsilon_n\in\{0,1\}$. The absolutely convergent series $\xi=\sum_{n\ge0}\varepsilon_n3^{-n}$ is real. A number is transcendental if it is not algebraic over $\mathbb Q$.

Statement recorded in the supplied source.

For every such sequence, does $\xi\notin\mathbb Q$ imply that $\xi$ is transcendental? [S2]

Short English statement

Must every irrational number formed from only zero and one ternary digits be transcendental?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700018 (AMR-086-0018). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 2.15. https://arxiv.org/abs/math/0312440.