The order of appearance of the product of two Fibonacci and Lucas numbers
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THE ORDER OF APPEARANCE OF THE PRODUCT OF TWO FIBONACCI AND LUCAS NUMBERS N. IRMAK1,∗ and P. K. RAY2 1
¨ Institute of Mathematics, Ni˘ gde Omer Halisdemir University, Ni˘ gde, Turkey e-mails: [email protected], [email protected] 2
Sambalpur University, Jyoti Vihar, Burla, India e-mail: [email protected]
(Received December 10, 2019; revised March 17, 2020; accepted March 29, 2020)
Abstract. Let Fn and Ln be the nth Fibonacci and Lucas number, respectively. The order of appearance is defined as the smallest natural number k such that n divides Fk and denoted by z(n). In this paper, we give explicit formulas for the terms z (Fa Fb ), z (La Lb ), z (Fa Lb ) and z (Fn Fn+p Fn+2p ) with p ≥ 3 prime.
1. Introduction Let (Fn )n≥0 be the Fibonacci sequence given by Fn+2 = Fn+1 + Fn , where F0 = 0 and F1 = 1. Let (Ln )n≥0 be the Lucas sequence satisfying the same recursive relationship as the Fibonacci sequence with initial conditions L0 = 2 and L1 = 1. These numbers have very amazing properties (for the details, refer to the book [1]). The study of the divisibility properties of Fibonacci and Lucas numbers is one of the popular topics in number theory. Let n be a positive integer, the order of appearance of n in Fibonacci sequence, denoted by z (n), is defined as the smallest positive integer k, such that n | Fk . There are many results about the z (n) in the literature. Lehmer [2] proved that all solutions of the equation z (n) = n ∓ 1 are prime numbers. Motivated by this result, Trojovsk´ y [10] solved the Diophantine equation z (n) = n ∓ l for |l| ∈ {1, . . . , 9}. Sall´e [9] proved that z (n) < 2n, for all positive integers n. Recently, Trojovsk´ a [11] solved the equation z (n) = (2 − 1/k)n for positive integers n and k. ∗ Corresponding
author. Key words and phrases: order of appearance, Fibonacci number, Lucas number. Mathematics Subject Classification: 11B39.
c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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N. N. IRMAK IRMAK and and P. P. K. K. RAY RAY
There are also several results including Fibonacci and Lucas numbers. For instance, z (Fn ) = n and z (Ln ) = 2n. Marques [4–6] established several formulas about the order of appearance of product of consecutive Fibonacci and Lucas numbers. Let En be the nth Fibonacci or Lucas number. The explicit formulas of the terms z (En En+1 ), z (En En+1 En+2 ), z (En+1 En+2 En+3 En+4 ) and z (Enk ) were given by Marques. Later, Pongsriiam [7] gave an algorithm to evaluate the term z (En+1 En+2 · · · En+k ). It is natural to ask: What are the formulas of the order of appearance of product of arbitrary two Fibonacci or Lucas numbers if they exist? In this article, in order to bridge this gap, we establish formulas of the terms z (Fa Fb ), z (La Lb ) and z (Fa Lb ). Moreover, we also find explicit formulas for the terms z(FnFn+p Fn+2p ). Our results are following: Theorem 1.1. Let a, b ≥ 3 be integers and d = gcd(a, b). Then we have (i) (ii)
(iii)
z (Fa Fb ) = [a, b]Fd , � [a,
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