Reprint: On the number of integers which can be represented by a binary form (1938)

  • Pál Erdős

  • Kurt Mahler


Let F(x,y)F(x,y) be a binary form of degree n3n\ge 3 with integer coefficients and non-vanishing discriminant, and let A(u)A(u) be the number of different positive integers kuk\le u, for which F(x,y)=k|F(x,y)|=k has at least one solution in integers x,yx,y. In this paper, using Mahler's pp-adic generalisation of the Thue-Siegel theorem, Erdős and Mahler prove that lim infuA(u)u2/n>0.\liminf_{u\to\infty} A(u)u^{-2/n}>0.

Reprint of the authors' paper [J. Lond. Math. Soc. 13, 134--139 (1938; Zbl 0018.34401; JFM 64.0116.01)].