On the Resolution of the Equations

The purpose of the present paper is to prove that there are finitely many binomial coefficients of the form (f in certain binary recurrences, and give a simple method for the determination of these coefficients. We illustrate the method by the Fibonacci, the Lucas, and the Pell sequences. First, we transform both of the title equations into two elliptic equations and apply a theorem of Mordell [10], [11] to them. (Later, Siegel [16] generalized MofdelTs result, and in 1968 Baker [1] gave its effective version.) After showing the finiteness, we use the program package SIMATH [15] which is a computer algebra system, especially useful for number theoretic purposes, and is able to find all the integer points on the corresponding elliptic curves. The algorithms of SIMATH are based on some deep results of Gebel, Petho, and Zimmer [5]. Before going into detail, we present a short historical survey. Several authors have investigated the occurrence of special figurate numbers in the second-order linear recurrences. One such problem is, for example, to determine which Fibonacci numbers are square. Cohn [2], [3] and Wyler [18], applying elementary methods, proved independently that the only square Fibonacci numbers are F0 = 0, Fl=F2 = l, and Fl2 = 144. A similar result for the Lucas numbers was obtained by Cohn [4]: if Ln = x, then n-\ or n = 3. London and Finkelstein [6] established Ml Fibonacci cubes. Petho [12] gave a new proof of the theorem of London and Finkelstein, applying the Gel'fond-Baker method and computer investigations. Later Petho found all the fifthpower Fibonacci numbers [14], and all the perfect powers in the Pell sequence [13]. Another special interest was to determine the triangular numbers Tx = ^ in certain recurrences. Hoggatt conjectured that there are only five triangular Fibonacci numbers. This problem was originally posed in 1963 by Tallman [17] in The Fibonacci Quarterly. In 1989 Mng [8] proved Hoggatt's conjecture by showing that the only Fibonacci numbers that are triangular are FQ 0, F'{ = F2 1, F4 = 3, F^ 2 1 , and Fl0 = 55. Ming also proved in [9] that the only triangular Lucas numbers are Lx 1, L2 3, and Ll% = 5778. Moreover, the only triangular Pell number is Px = 1 (see McDaniel [7]). Since the number Tx-\ is equal to the binomial coefficient (2), it is natural to ask whether the terms (3) occur in binary recurrences or not. As we will see, the second-order linear recurrences, for instance, the Fibonacci, the Lucas, and the Pell sequences have few such terms. Now we introduce some notation. Let the sequence {Un}TM=0 be defined by the initial terms U0, Ul9 and by the recurrence relation U„ = AUn_l+BUn_2 (»>2), (1)