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

  • Kurt Mahler


The aim of this paper is to determine an explicit lower bound, free of unknown constants, for the distance of π\pi from a given rational or algebraic number. In particular, Mahler proves that, for all positive integers p,q2p,q\ge 2,

\par\parπpq>1q42.\par\par\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)].