This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker's theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented.

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Abstract. The diophantine equation x -3xy -y = ±3"° 17"' 19"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 -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.