Integer Points on Elliptic Curves

This chapter begins with preliminaries on Diophantine equations and Diophantine approximations, which are used later in the chapter. That includes Pell’s and Pellian equations, linear forms in logarithms and simultaneous rational approximations. We explain general methods for finding integer points on elliptic curves (transformation to Thue equations, application of elliptic logarithms, solving systems of Pellian equations via linear forms in logarithms and the Baker-Davenport reduction). These methods are then applied to the problem of finding all integer points on elliptic curves induced by Diophantine triples. We present the proof of an absolute upper bound for the size of Diophantine tuples and sketch the main steps in the results that lead to the proof of the non-existence of Diophantine quintuples.