Reprint: An arithmetic property of Taylor coefficients of rational functions (1935)

Abstract

Herein, Mahler shows that, if $R(z)={\sum }_{n\ge 0}G(n)z^n$ is a rational function having algebraic coefficients, infinitely many of which are zero, then there is a natural number $r$ and at most $r$ non-negative rational integers $r_1,r_2,\dots ,r_{\varrho }$, pairwise incongruent modulo $r$, such that only finitely many $G(n)$, with $n\equiv r_{\tau }(\mathrm{mod}r)$ and $n\ge r_{\tau }$ for $\tau =1,2,\dots ,\varrho$, vanish.

Reprint of the author's paper [Proc. Akad. Wet. Amsterdam 38, 50--60 (1935; Zbl 0010.39006; JFM 61.0176.02)].