The Number of Solutions of Diophantine Equations

0. Introduction. In two recent papers 4], 30], Erdd os, Stewart and the author showed that certain diophantine equations have many solutions. In this way they indicated how far certain results are capable for improvements at most. First we mention some relevant results from the literature on upper bounds for the numbers of solutions of diophantine equations and then we sketch how our method leads to opposite results. ja j j. Let m 2 Z be non-zero. In 1908 Thue 31] proved that, for irreducible polynomials f, the equation (1) f(x; y) = m has only nitely many solutions in x; y 2 Z and in 1929 Siegel 26] showed that there exists an explicit upper bound for the number of solutions of (1). An equation of type (1) is said to be a Thue equation. Thue's result was extended by Mahler in 1933. Let p 1 ; :::; p s be distinct prime numbers. Consider the equation Z 0 with gcd (x; y) = 1: Mahler 17] proved that, for irreducible f, equation (2) has at most c s+1 1 solutions where c 1 is some number depending only on f. An equation of type (2) is said to be a Thue-Mahler equation. Without much trouble the condition 'f is irreducible' can be relaxed to 'f has at least three distinct linear factors in its factorisation over the complex numbers'. Here we list some out of several upper bounds for the numbers of solutions of Thue and Thue-Mahler equations. In 1955 Davenport and Roth 3] proved that, for irreducible f, equation (1) has at most (4A) 2n 2 j m j 3 + e 643n 2 solutions (x; y): Lewis and Mahler 15] showed that, for f with non-zero discriminant, equation (2) has at most