The Pell Equation x 2 − ( k 2 − k ) y 2 = 2 t Ahmet

Let k, t, d be arbitrary integers with k ≥ 2, t ≥ 0 and d = k − k. In the first section we give some preliminaries from Pell equations x − dy = 1 and x − dy = N , where N be any fixed positive integer. In the second section, we consider the integer solutions of Pell equations x − dy = 1 and x − dy = 2. We give a method for the solutions of these equations. Further we derive recurrence relations on the solutions of these equations. Keywords—Pell equation, solutions of Pell equation. I. PRELIMINARY FACTS. Let d = 1 be a positive non-square integer and N be any fixed positive integer. Then the equation x − dy = ±N (1) is known as Pell equation and is named after John Pell (16111685), a mathematician who searched for integer solutions to equations of this type in the seventeenth century. Ironically, Pell was not the first to work on this problem, nor did he contribute to our knowledge for solving it. Euler (17071783), who brought us the ψ-function, accidentally named the equation after Pell, and the name stuck. For N = 1, the Pell equation x − dy = ±1 (2) is known as the classical Pell equation and was first studied by Brahmagupta (598-670) and Bhaskara (1114-1185), (see [1]). Its complete theory was worked out by Lagrange (1736-1813), not Pell. It is often said that Euler (1707-1783) mistakenly attributed Brouncker’s (1620-1684) work on this equation to Pell. However the equation appears in a book by Rahn (16221676) which was certainly written with Pell’s help: some say entirely written by Pell. Perhaps Euler knew what he was doing in naming the equation. Baltus [2], Kaplan and Williams [5], Lenstra [7], Matthews [8], Mollin, Poorten and Williams [9], Stevenhagen [10], Tekcan [12,13,14], and the others consider some specific Pell equations and their integer solutions. Further details on Pell equations can be found in [3,10]. The Pell equation in (2) has infinitely many integer solutions (xn, yn) for n ≥ 1. The first non-trivial positive integer solution (x1, y1) (in this case x1 or x1+y1 √ d is minimum) of this equation is called the fundamental solution, because all other solutions can be (easily) derived from it. In fact, if (x1, y1) is the fundamental solution of x − dy = 1, then the n-th positive solution of it, say (xn, yn), is defined by the equality xn + yn √ d = (x1 + y1 √ d) (3) Ahmet Tekcan is with the Uludag University, Department of Mathematics, Faculty of Science, Bursa-TURKEY, email: [email protected], http://matematik.uludag.edu.tr/AhmetTekcan.htm. for integer n ≥ 2. (Furthermore, all nontrivial solutions can be obtained considering the four cases (±xn,±yn) for n ≥ 1.) There are several methods for finding the fundamental solution of Pell’s equation x − dy = 1 for a positive nonsquare integer d, e.g., the cyclic method [4, p. 30], known in India in the 12-th century, or the slightly less efficient but more regular English method (17-th century) which produce all solutions of x2−dy2 = 1 [4, p. 32]. But the most efficient method for finding the fundamental solution is based on the simple finite continued fraction expansion of √ d. We can describe it as follows (see [2] and also [6, p.154]): Let [a0; a1, a2, · · · , ar, 2a0] be the simple continued fraction of √ d, where a0 = √ d . Let p0 = a0, p1 = 1 + a0a1, q0 = 1, q1 = a1. In general pn = anpn−1 + pn−2 (4) qn = anqn−1 + qn−2 for n ≥ 2. Then the fundamental solution of x − dy = 1 is (x1, y1) = ⎧⎨ ⎩ (pr, qr) if r is odd (p2r+1, q2r+1) if r is even. (5) On the other hand, in connection with (1) and (2), it is well known that if (X1, Y1) and (xn−1, yn−1) are integer solutions of x − dy = ±N and x − dy = 1, respectively, then (Xn, Yn) is also a positive solution of x2−dy2 = ±N , where Xn + dYn = (xn−1 + dyn−1)(X1 + dY1) (6) for n ≥ 2. In this work we will define by recurrence an infinite sequence of positive solutions of the Pell equation x2−dy2 = 2, where d = k − k with k ≥ 2 an integer and t ≥ 0 is also an integer. We will also express the obtained solutions for t ≥ 1 in terms of the “fundamental solution” of x2−dy2 = 1 in two cases k = 2 or k ≥ 3. II. THE PELL EQUATION x − (k − k)y = 2 . Let d = k − k be a positive non-square integer for an integer k ≥ 2 and let t ≥ 0 be an arbitrary integer. In this section we consider the integer solutions of Pell equation x2− (k − k)y = 2. First we consider the case t = 0, that is, the classical Pell equation x − (k − k)y = 1. Theorem 2.1: Let d = k − k with k ≥ 2. Then 1) The continued fraction expansion of √ d is given by √ d = ⎧⎨ ⎩ [1; 2] if k = 2 [k − 1; 2, 2k − 2] otherwise. World Academy of Science, Engineering and Technology 43 2008