Reprint: On the approximation of $\pi$ (1953)

Abstract

The aim of this paper is to determine an explicit lower bound, free of unknown constants, for the distance of $\pi$ from a given rational or algebraic number. In particular, Mahler proves that, for all positive integers $p,q\ge 2$,

\( \par \par \left|\pi-\frac{p}{q}\right|>\frac{1}{q^{42}}. \par \par \)

Reprint of the author's paper [Nederl. Akad. Wet., Proc., Ser. A 56 (Indag. Math. 15), 30--42 (1953; Zbl 0053.36105)].