Reprint: On the lattice points on curves of genus $1$ (1935)

Abstract

Let $F(x,y)$ be a cubic binary form with integer coefficients that is irreducible over the field of rational numbers, and let $k\neq 0$ be an integer. Further, let $A(k)$ be the number of pairs of integers $(x,y)$ satisfying $F(x,y)=k$. Here, Mahler proves that $A(k)$ is unbounded, and that there are infinitely many integers $k$ such that $A(k)\geqslant \sqrt[4]{\log k}.$

Reprint of the author's paper [Proc. Lond. Math. Soc. (2) 39, 431--466 (1935; Zbl 0012.15006; JFM 61.0146.02); corrigendum ibid. 40, 558 (1936; JFM 61.1055.02)].