Solving a specific Thue-Mahler equation

Abstract

The diophantine equation x 3 − 3 x y 2 − y 3 = ± 3 n 0 17 n 1 19 n 2 {x^3} - 3x{y^2} - {y^3} = \pm {3^{{n_0}}}{17^{{n_1}}}{19^{{n_2}}} is completely solved as follows. First, a large upper bound for the variables is obtained from the theory of linear forms in p-adic and real logarithms of algebraic numbers. Then this bound is reduced to a manageable size by p-adic and real computational diophantine approximation, based on the L 3 {L^3} -algorithm. Finally the complete list of solutions is found in a sieving process. The method is in principle applicable to any Thue-Mahler equation, as the authors will show in a forthcoming paper.