Verification of a conjecture of E. Thomas
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 \textit{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 [\textit{E. Thomas}, Solutions to certain families of Thue equations, J. Number Theory 43, No. 3, 319--369 (1993; Zbl 0774.11013)].
sharp lower bounds for linear forms in two logarithms
scientific article; zbMATH DE number 431318