On the Lucas sequence equation $$\frac{1}{U_n}=\sum _{k=1}^{\infty }\frac{U_{k-1}}{x^k}$$
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On the Lucas sequence equation
1 Un
=
∞
Uk−1 k=1 x k
Szabolcs Tengely1
© Akadémiai Kiadó, Budapest, Hungary 2015
Abstract In 1953 Stancliff noted an interesting property of the Fibonacci number F11 = 89. One has that 1 1 2 3 5 0 1 = + 2 + 3 + 4 + 5 + 6 + ..., 89 10 10 10 10 10 10 where in the numerators the elements of the Fibonacci sequence appear. We provide methods to determine similar identities in case of Lucas sequences. As an example we prove that ∞
Uk 1 1 = = , U10 416020 647k+1 k=0
where U0 = 0, U1 = 1 and Un = 4Un−1 + Un−2 , n ≥ 2. Keywords
Lucas sequences · Diophantine equations · Elliptic curves
Mathematics Subject Classification
Primary 11D25 · Secondary 11B39
1 Introduction Stancliff [16] noted without proof an interesting property of the Fibonacci sequence Fn . One has that ∞
Fk 1 1 = 0.0112358 . . . = = . F11 89 10k+1 k=0
B 1
Szabolcs Tengely [email protected] Mathematical Institute, University of Derecen, P.O.Box 12, Debrecen 4010, Hungary
123
Sz. Tengely
In 1980 Winans [23] investigated the related sums ∞ Fαk 10k+1 k=0
for certain values of α. In 1981 Hudson and Winans [8] provided a complete characterization of all decimal fractions that can be approximated by sums of the type n 1 Fαk , α, l ≥ 1. Fα 10l(k+1) k=1
Long [12] proved a general identity for binary recurrence sequences from which one obtains e.g. ∞
Fk 1 = , 9899 102(k+1) k=0
∞
1 Fk = . 109 (−10)k+1 k=0
In the previous examples decimal fractions were studied, in case of different bases characterizations were obtained by Jia Sheng Lee [10] and by Köhler [9] and by Jin Zai Lee and Jia Sheng Lee [11]. Here we state a result by Köhler that we will use later in this article. Theorem 1.1 Let A, B, a0 , a1 be arbitrary complex numbers. Define the sequence {an } by the recursion an+1 = Aan + Ban−1 . Then the formula ∞ ak a0 x − Aa0 + a1 = 2 k+1 x x − Ax − B k=0
holds for all complex x such that |x| is larger than the absolute values of the zeros of x 2 − Ax − B. Let P and Q be non-zero relatively prime integers. The Lucas sequence {Un (P, Q)} is defined by U0 = 0, U1 = 1
Un = PUn−1 − QUn−2 ,
and
if n ≥ 2.
In this paper we deal with the determination of all integers x ≥ 2 for which there exists an n ≥ 0 such that ∞ 1 Uk−1 = , (1.1) Un xk k=1
where Un is a Lucas sequence with some given P and Q. In case of P = 1, Q = −1 one gets the Fibonacci sequence. De Weger [5] computed all x ≥ 2 in case of the Fibonacci sequence, the solutions are as follows ∞
1 1 1 Fk−1 = = = , F1 F2 1 2k k=1
1 1 = = F10 55
∞ k=1
Fk−1 , 8k
∞
1 1 Fk−1 = = , F5 5 3k
1 1 = = F11 89
k=1 ∞ Fk−1 . 10k k=1
De Weger applied arguments of algebraic number theory and obtained two Thue equations, which were solved using Baker’s method (see e.g. [2,7,15]). In the current work, we show
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On the Lucas sequence equation...
Uk−1 how to reduce a search for integral x ≥ 2 related to the equation U1n = ∞ k=1 x k to elliptic Diophantine equations or to Thue equations following an elementary argument by Alekseyev an
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