Some Special Pell Equations

Let D be a positive non-square integer and let (t, u) denote the fundamental solution of the Pell equation

$$T^2 - DU^2 = 1$$

((6.1))

. In this chapter we will be concerned with Pell equations for which the values of t and u tend to be small. As we shall see later in Chapter 9, this appears to be a very unusual circumstance; for this reason, then, if no other, it is of some interest to investigate this phenomenon. For example, consider the case of D = M² − 1 for $M \in \mathbb{Z}^{>0}$. Clearly, (M, 1) is a solution of the Pell equation (6.1). Furthermore, it is easy to verify that this is the fundamental solution. Of course, we might regard such Pell equations as being easy to solve, and in a sense this is the case; however, there are a few problems that do develop.