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 \[A(k)\geqslant \sqrt[4]{\log k}.\] \par 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)].

11-03, 11D41, 11G99, 14H25