Let be a cubic binary form with integer coefficients that is irreducible over the field of rational numbers, and let be an integer. Further, let be the number of pairs of integers satisfying . Here, Mahler proves that is unbounded, and that there are infinitely many integers such that
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)].