The x -coordinates of Pell equations and sums of two Fibonacci numbers II
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The x-coordinates of Pell equations and sums of two Fibonacci numbers II MAHADI DDAMULIRA1,∗
and FLORIAN LUCA2,3,4
1 Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24/II, 8010 Graz, Austria 2 School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa 3 Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia 4 Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico *Corresponding author. Email: [email protected]; [email protected]; [email protected]
MS received 6 August 2019; accepted 28 January 2020 Abstract. Let {Fn }n≥0 be the sequence of Fibonacci numbers defined by F0 = 0, F1 = 1 and Fn+2 = Fn+1 + Fn for all n ≥ 0. In this paper, for an integer d ≥ 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − dy 2 = ±4 which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize. Keywords. method.
Fibonacci number; Pell equation; linear form in logarithm; reduction
Mathematics Subject Classification.
11B39, 11D45, 11D61, 11J86.
1. Introduction Let {Fn }n≥0 be the sequence of Fibonacci numbers given by F0 = 0, F1 = 1
and
Fn+2 = Fn+1 + Fn for all n ≥ 0.
The Fibonacci sequence is sequence A000045 on the On-Line Encyclopedia of Integer Sequences (OEIS). The first few terms of this sequence are {Fn }n≥0 = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, . . . . In this paper, we let U := {Fn + Fm : n ≥ m ≥ 0} be the sequence of sums of two Fibonacci numbers. The first few members of U are U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, 22, 23, 24, 26, . . .}. Let d ≥ 2 be a positive integer which is not a square. It is well known that the Pell equation x 2 − dy 2 = ±4,
(1)
© Indian Academy of Sciences 0123456789().: V,-vol
58
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Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:58
has infinitely many positive integer solutions (x, y). By putting (x1 , y1 ) for the smallest positive solutions to (1), all solutions are of the forms (xk , yk ) for some positive integer k, where √ √ k x1 + y1 d xk + yk d = for all k ≥ 1. 2 2 Furthermore, the sequence {xk }k≥1 is binary recurrent. In fact, the following formula √ k √ k x1 + y1 d x1 − y1 d xk = + . 2 2 holds for all positive integers k. Recently, Gómez and Luca [9] studied the Diophantine equation xk = Fm + Fn , with n ≥ m ≥ 0,
(2)
where xk are the x-coordinates of the solutions of the Pell equation x 2 − dy 2 = ±1 for some positive integer k and {Fn }n≥0 is the sequence of Fibonacci numbers. They proved that for each square free integer d ≥ 2, there is at most one positive integer k such that xk admits the representation (3) for some nonnegative integers 0 ≤ m ≤ n, except for d ∈ {2, 3, 5, 11, 30}. Furthermore, they explicitly stated all the solutions for these exceptional cases. In the same spirit, Bravo et al. [1] studied the Diophantine equation xk = Tm + Tn ,
with n ≥
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