where A1, A2 are non-zero integers, D1, D2 are non-square positive integers and A1D2−A2D1 6= 0. Note that, if the last condition is not satisfied, then A2/A1 = D2/D1 = q 2 for some q ∈ Q and the solutions (X, Y, Z) of the system of equations are given by (X, Y, qX), where (X, Y ) is a solution of the first equation of the system. A large number of papers is devoted to the study of simultaneous Pell equations; they could be classified into three categories: The first one includes papers, in which results are proved of the type “there are no solutions”, or “there are at most k solutions”, where k is a number between, say, 1 and 4; such are the papers [4],[5],[6],[7],[8],[20],[21], [26],[38],[39],[40]. The generality of such results is obtained at the cost of restricting at least two parameters among A1, D1, A2, D2. Of course, if a system (1) falls within the scope of such a paper and 3 one is so lucky that ∗Department of Mathematics, University of Crete, Iraklion, Greece, e-mail: tzanakis@math.uoc.gr , http://www.math.uoc.gr/ tzanakis Throughout this paper the term “algorithm”, in a broad sense though, could also be used in place of the word “method”. Often, solving (1) is reduced to finding common terms of two distinct second order recurrence sequences, a problem studied in a number of papers without any direct reference to (1); in our bibliography it is natural to include such papers. This is the very optimistic case!