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

### Kurt Mahler

## Abstract

In 1908, Thue showed that, if $ζ$ is a real algebraic number of degree $n$ and $Θ$ is a positive number, the inequality $ qp −ζ ≤q_{−(2n+1+Θ)}$ has only finitely many rational solutions $p/q$. Siegel, in 1920, showed that one can replace the exponent $2n +1+Θ$ by $β=min_{1≤s≤n−1}(s+1n +s+Θ).$

In this article, Mahler establishes the following $p$-adic extension of Siegel's result. Let $f(x)$ be an irreducible polynomial with rational integer coefficients of degree $n≥3$, let $P_{1},P_{2},…,P_{t}$ be finitely many prime numbers and let $ζ,ζ_{1},ζ_{2},…,ζ_{t}$ be real zero of $f(x)$, a $P_{1}$-adic zero of $f(x)$, a $P_{2}$-adic zero of $f(x),…$, and a $P_{t}$-adic zero of $f(x)$, respectively. Mahler proves that, if $β$ is Siegel's exponent and $k≥1$ 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 $p/q$.

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].