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Mathematics of Computation

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Solving a specific Thue-Mahler equation

Authors: N. Tzanakis and B. M. M. de Weger
Journal: Math. Comp. 57 (1991), 799-815
MSC: Primary 11D25; Secondary 11D61, 11Y50
MathSciNet review: 1094961
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Abstract: The diophantine equation $ {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}$-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.

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Article copyright: © Copyright 1991 American Mathematical Society

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