Erdős, Pál; Mahler, Kurt

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

Doc. Math. Extra Vol. Mahler Selecta, 475-481 (2019)
DOI: 10.25537/dm.2019.SB-475-481


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 \[\liminf_{u\to\infty} A(u)u^{-2/n}>0.\] \par Reprint of the authors' paper [J. Lond. Math. Soc. 13, 134--139 (1938; Zbl 0018.34401; JFM 64.0116.01)].

Mathematics Subject Classification

11-03, 11E16, 11D85


Erdős, Pál
Mahler, Kurt