Rational approximation to algebraic numbers of small height: The Diophantine equation \(|ax^n-by^n|=1\) (Q2720955)

The main tool of this long and important work is the \textit{multidimensional hypergeometric method} for rational and algebraic approximation of certain algebraic numbers, which was considered first by K.~Mahler and sharpened by G.~Chudnovsky. Here the author obtains explicit results. His most spectacular result is the following definitive NEWLINENEWLINENEWLINETheorem. Let \(a\), \(b\) and \(n\) be given positive rational integers with \(n\geq 3\). Then the Thue equation NEWLINE\[NEWLINE |a x^n-by^n|=1 NEWLINE\]NEWLINE posseses at most one solution in positive rational integers \(x\) and \(y\). NEWLINENEWLINENEWLINEThis result is obtained as a corollary on new results on explicit Padé approximations to systems of binomial functions (studied both analytically and arithmetically). The proof also contains applications of new Chebyshev-like explicit estimates for primes in arithmetical progressions, it involves also sharp computational techniques and needed a lot of computer time. NEWLINENEWLINENEWLINEIndeed the main theorem of the paper is an explicit measure of irrationality for numbers of the type \((b/a)^{1/n}\) (Theorem~7.1), several corollaries are given for various Diophantine equations. NEWLINENEWLINENEWLINENo doubt that the very precise results proved in this paper will be used in other applications to Diophantine equations.