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

Abstract

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

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