Combinatorial numbers in binary recurrences
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COMBINATORIAL NUMBERS IN BINARY RECURRENCES ¨nde Kova ´cs Tu [Communicated by Attila Peth˝ o] University of Debrecen, Institute of Mathematics Debrecen, P.O. Box 12., H-4010 Hungary E-mail: [email protected] (Received July 24, 2008; Accepted October 31, 2008)
Abstract We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.
1. Introduction There are many papers about values of a polynomial p(x) ∈ Q[x] (taken at integer values of x) in a binary linear recurrence sequence U . The first such results dealt with the case where U is a special sequence and p(x) = xm with some m ≥ 2. That is, we are interested in terms of U which are perfect powers. In 1962 Ogilvy [22], one year later Moser and Carlitz [20], and Rollett [30] proposed the following problem: determine all squares in the Fibonacci sequence F . The problem was solved by Cohn [7], [8] and Wyler [39] who independently proved with elementary methods that the only squares in the Fibonacci sequence are F0 = 0, F1 = F2 = 1, F12 = 144. Later, Alfred [1] and Cohn [9] determined the squares in the Lucas sequence L. Peth˝o [25] and Cohn [10] independently determined the perfect powers in the Pell sequence. Recently, Bugeaud, Mignotte and Siksek [6] Mathematics subject classification numbers: 11B37, 11B83, 11Y50. Key words and phrases: binary recurrence sequences, polynomial values, combinatorial numbers. 0031-5303/2009/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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showed that the perfect powers in the Fibonacci and Lucas sequences are exactly F0 = 0, F1 = F2 = 1, F6 = 8, F12 = 144, and L1 = 1, L3 = 4, respectively. Another branch of problems is about triangular numbers in recurrence sequences, i.e. we take the polynomial p(x) = x(x+1) . Hoggatt stated the conjecture 2 that there are only five triangular Fibonacci numbers. In 1989 Ming [18] proved that this conjecture is true. Furthermore, Ming [19] and McDaniel [17] determined the triangular numbers in the Lucas and Pell sequences, respectively. In [34] Szalay described all values of the polynomials S2 (x) and S3 (x) in the Fibonacci, Lucas and Pell sequences, where Sk (x) denotes the sum of kth powers up to x − 1. Further, he listed all numbers of the form x4 in the Fibonacci and Lucas sequences, as well. As a generalization of the previous results, Tengely [37] recently determined the g-gonal numbers in t
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