Mahler, Kurt

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

Doc. Math. Extra Vol. Mahler Selecta, 557-570 (2019)
DOI: 10.25537/dm.2019.SB-557-570


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 \[ \par \par \left|\pi-\frac{p}{q}\right|>\frac{1}{q^{42}}. \par \par \] \par Reprint of the author's paper [Nederl. Akad. Wet., Proc., Ser. A 56 (Indag. Math. 15), 30--42 (1953; Zbl 0053.36105)].

Mathematics Subject Classification

11-03, 11J81