Reprint: On the approximation of algebraic numbers. I: On the greatest prime divisor of binary forms (1933)

  • Kurt Mahler


In 1908, Thue showed that, if ζ\zeta is a real algebraic number of degree nn and Θ\Theta is a positive number, the inequality pqζq(n2+1+Θ)\left|\frac{p}{q}-\zeta\right|\le q^{-\left(\frac{n}{2}+1+\Theta\right)} has only finitely many rational solutions p/qp/q. Siegel, in 1920, showed that one can replace the exponent n2+1+Θ\frac{n}{2}+1+\Theta by β=min1sn1(ns+1+s+Θ).\beta=\min_{1\le s\le n-1}\left(\frac{n}{s+1}+s+\Theta\right).

In this article, Mahler establishes the following pp-adic extension of Siegel's result. Let f(x)f(x) be an irreducible polynomial with rational integer coefficients of degree n3n\ge 3, let P1,P2,,PtP_1,P_2,\ldots,P_t be finitely many prime numbers and let ζ,ζ1,ζ2,,ζt\zeta,\zeta_1,\zeta_2,\ldots,\zeta_t be real zero of f(x)f(x), a P1P_1-adic zero of f(x)f(x), a P2P_2-adic zero of f(x),f(x), \ldots, and a PtP_t-adic zero of f(x)f(x), respectively. Mahler proves that, if β\beta is Siegel's exponent and k1k\ge 1 is fixed, the inequality \min\left{1,\left|\tfrac{p}{q}-\zeta\right|\right\} \prod_{\tau=1}^t \min{1,|p-q\zeta_\tau|_{P_\tau}\}\le k\, \max{|p|,|q|\}^{-\beta} has finitely many solutions in reduced rational numbers p/qp/q.

Reprint of the author's paper [Math. Ann. 107, 691--730 (1933; Zbl 0006.10502; JFM 59.0220.01)]. For Part II see [Zbl 1465.11013].