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On the practical solution of Thue-Mahler equations - an outline. (English) Zbl 0703.11014
Number theory, Vol. II. Diophantine and algebraic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 1037-1050 (1990).
[For the entire collection see Zbl 0694.00006.]
The authors have given [J. Number Theory 31, No.2, 99-132 (1989; Zbl 0657.10014)] a marvelous algorithm to determine the solutions of \(f(x,y)=m\), x,y,m\(\in {\mathbb{Z}}\), m constant, \(f\in {\mathbb{Z}}[X,Y]\) an irreducible form of degree at least three, which lowers the bounds given by Baker’s estimates of linear forms in logarithms by powers of e.
In the first paper, Tzanakis presents a survey about this method, giving explicitly all integer solutions of \[ X^ 4-4X^ 3Y-12X^ 2Y^ 2+4Y^ 4=1,\quad X^ 4-12X^ 2Y^ 2-8XY^ 3-4Y^ 4=1 \] to find all (22) integral points on the elliptic curve \(y^ 2=x^ 3-4x+1.\)
In the second paper, de Weger takes advantage of Yu’s methods of linear forms in p-adic logarithms and the Lenstra-Lenstra-Lovász algorithm to obtain equally far-reaching results upon the Thue-Mahler equations \(f(x,y)=\prod_{i}p_ i^{a_ i}\) (everything as above, but now only a finite number of primes is prescribed, not the exponents). All (26) solutions of \[ X^ 3-3XY^ 2-Y^ 3=\pm 3^{n_ 1}17^{n_ 2}19^{n_ 3} \] with \(x>0\), \(y\geq 0\), \((x,y)=1\) are described (as in detail considered by both authors in “Solving a specific Thue-Mahler equation” [Memorandum No.793, Twente University, 1989]).
Reviewer: B.Richter

MSC:
11D41 Higher degree equations; Fermat’s equation