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How to explicitly solve a Thue-Mahler equation. (English) Zbl 0773.11023
Let \(F(X,Y)\in\mathbb{Z}[X,Y]\) be a homogeneous, irreducible polynomial of degree \(n\geq 3\); \(p_ 1,\dots,p_ v\) be distinct rational primes and \(c\in\mathbb{Z}\). The aim of this paper is to present a practical method for the resolution of the Thue-Mahler equation \[ F(X,Y)=cp_ 1^{z_ 1}\cdot\dots\cdot p^{z_ v}_ v \] in \((X,Y,z_ 1,\dots,z_ v)\in\mathbb{Z}^ 2\times\mathbb{Z}^ v_{\geq 0}\) under the natural assumption \((X,Y)=1\). The method is based on the combination of lower bounds for linear forms in complex and \(p\)-adic logarithms of algebraic numbers with numerical diophantine approximation techniques.
Although the basic ideas are the same as in [Math. Comput. 57, 799-815 (1991; Zbl 0738.11029)] by the same authors, there are involved in the new method considerable improvements. The first is “The Prime Ideal Removing Lemma” in section 5, which makes it possible to reduce the number of linear forms to be analysed. The second is the combination of the LLL-reduction steps with the Fincke-Pohst method in section 17. In section 18 a new sieving procedure is described for the localization of small solutions of \(S\)-unit equations.
The method is demonstrated on the resolution of the equation \[ x^ 3- 23x^ 2y+5xy^ 2+24y^ 3=\pm 2^{z_ 1}3^{z_ 2}5^{z_ 3}7^{z_ 4}. \] It is proved that this equation has 72 solutions with \(x\geq 0\).

MSC:
11D57 Multiplicative and norm form equations
11Y50 Computer solution of Diophantine equations
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