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 is a real algebraic number of degree and is a positive number, the inequality has only finitely many rational solutions . Siegel, in 1920, showed that one can replace the exponent by

In this article, Mahler establishes the following -adic extension of Siegel's result. Let be an irreducible polynomial with rational integer coefficients of degree , let be finitely many prime numbers and let be real zero of , a -adic zero of , a -adic zero of , and a -adic zero of , respectively. Mahler proves that, if is Siegel's exponent and 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 .

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].