Simultaneous Pell equations

Let R and S be positive integers with R < S. We shall call the simultaneous Diophantine equations x −Ry = 1, z − Sy = 1 simultaneous Pell equations in R and S. Each such pair has the trivial solution (1, 0, 1) but some pairs have nontrivial solutions too. For example, if R = 11 and S = 56, then (199, 60, 449) is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible to show that the number of solutions is always finite, and it is possible to give a complete list of them. In this paper we report on the solutions when R < S ≤ 200. Let R and S be positive integers with R < S. We shall call the simultaneous Diophantine equations x −Ry = 1, z − Sy = 1 simultaneous Pell equations in R and S. Each such pair has the trivial solution (1, 0, 1) but some pairs have nontrivial solutions too. For example, if R = 11 and S = 56, then (199, 60, 449) is a solution. Indeed, there are infinitely many simultaneous Pell equations with nontrivial solutions, as can be seen by taking y = 2, R = k + k, and S = m + m. Using theorems due to Siegel [4, §1] and Baker [1], it is possible to show that the number of solutions is always finite, and it is possible to give a complete list of them. This is exactly what we have done, for all 19,900 simultaneous Pell equations with R < S ≤ 200, and this paper, precisely, is a report on our method and results. Note that the term ‘simultaneous Pell equations’ could be defined to apply to other pairs, such as x −Ry = 1, z − Sx = 1. There are other variants as well, including the ‘simultaneous Pellian equations’ solved in an article by R. G. E. Pinch [3] which overlaps this paper to some extent. All these ‘simultaneous Pells’ can be solved using methods similar to those described here. Some of the simultaneous Pell equations under consideration in this paper can be solved simply by factoring. This is the case if R or S or RS is a square. In the Received by the editor June 8, 1994 and, in revised form, October 11, 1994. 1991 Mathematics Subject Classification. Primary 11D09.