Conjecture 5.4 — Loxton and van der Poorten

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 $0<n_0<n_1<\cdots$ be integers, $p$ a prime, and let $\mathbb F_p$ be the field of $p$ elements. Set $f(z)=\sum_{i\ge0}z^{n_i}\in\mathbb F_p[[z]]$ and $\xi=\sum_{i\ge0}10^{-n_i}\in\mathbb R$. Algebraicity of $f$ over $\mathbb F_p(z)$ means that a nonzero polynomial $P(z,T)\in\mathbb F_p[z,T]$ satisfies $P(z,f(z))=0$; irrationality means $f\notin\mathbb F_p(z)$.

Statement recorded in the supplied source.

If for some prime $p$ the series $f$ is algebraic but irrational over $\mathbb F_p(z)$, must $\xi$ be transcendental over $\mathbb Q$? [S2]

Short English statement

Does an algebraic but nonrational generating series in some finite characteristic force the corresponding decimal number with digits at those positions to be transcendental?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700067 (AMR-086-0067). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 5.4, p.48. https://arxiv.org/abs/math/0312440v2.