Rational approximation to algebraic numbers of small height: the Diophantine equation |ax~n - by~n| = 1

摘要

Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional "hypergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a, b and n are given positive integers with n ≧ 3, we show that the equation of the title possesses at most one solution in positive integers x, y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pade approximations to systems of binomial functions, together with new Chebyshev-like estimates for primes in arithmetic progressions and a variety of computational techniques.