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Verification of a conjecture of E. Thomas. (English) Zbl 0780.11013

For each integer \(n>3\), the Diophantine equation \(X^3 - (n-1) X^2 - (n+2) XY^2 - Y^3=\pm 1\) in integers \(X\) and \(Y\) has only the six trivial solutions \(\pm (0,1)\), \(\pm (1,0)\), \(\pm(1,-1)\). The ranges \(n\geq 1.365\cdot 10^7\) and \(3<n \leq 10^3\) had been covered by E. Thomas [J. Number Theory 34, 235–250 (1990; Zbl 0697.10011)]. In the present paper, the author completes the proof by means of sharp lower bounds for linear forms in two logarithms; this is an example where refined numerical constants for such lower bounds play a fundamental role, because the proof involves a large number of linear forms in two logarithms.
This solution of Thomas’ conjecture provides the first example of a family of diophantine equations which can be solved by transcendence methods. Further examples have been produced more recently [E. Thomas, Solutions to certain families of Thue equations, J. Number Theory 43, No. 3, 319–369 (1993; Zbl 0774.11013)].

MSC:

11D25 Cubic and quartic Diophantine equations
11D59 Thue-Mahler equations
11J82 Measures of irrationality and of transcendence
11J86 Linear forms in logarithms; Baker’s method
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