Some Fundamental Methods in the Theory of Diophantine Equations

Abstract

Publisher Summary This chapter discusses some fundamental methods in the theory of diophantine equations. The study of diophantine equations amounts to the search for solutions in natural numbers, or rational numbers of polynomial or exponential equations. An equation may have some solutions that are obvious at first sight-these are called the trivial solutions, and the investigation always refers to the non-trivial solutions. The chapter discusses the three possibilities: (1) the equation has no solution (except the trivial solutions), (2) the equation has finitely many solutions, and (3) the equation has infinitely many solutions. Some recent methods, which involve algebraic geometry, real or complex analysis are also discussed. The birational equivalence classes of elliptic curves are classified by the orbits of the upper half plane H under the action of the group of unimodular transformations. An important and powerful method to study certain classes of diophantine equations is based on the determination of effective lower bounds for linear forms in logarithms.