Tribonacci numbers that are concatenations of two repdigits
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Tribonacci numbers that are concatenations of two repdigits Mahadi Ddamulira1 Received: 6 May 2020 / Accepted: 4 September 2020 / Published online: 15 September 2020 © The Author(s) 2020
Abstract Let (Tn )n≥0 be the sequence of Tribonacci numbers defined by T0 = 0, T1 = T2 = 1, and Tn+3 = Tn+2 + Tn+1 + Tn for all n ≥ 0. In this note, we use of lower bounds for linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction procedure to find all Tribonacci numbers that are concatenations of two repdigits. Keywords Tribonacci number · repdigit · linear form in logarithms · reduction method Mathematics Subject Classification Primary 11B39 · 11D45 · Secondary 11D61 · 11J86
1 Introduction A repdigit is a positive integer R that has only one distinct digit when written in its decimal expansion. That is, R is of the form R = d · · · d = d times
10 − 1 , 9
(1.1)
for some positive integers d, with ≥ 1 and 0 ≤ d ≤ 9. The sequence of repdigits is sequence A010785 on the On-Line Encyclopedia of Integer Sequences (OEIS) [8]. Consider the sequence (Tn )n≥0 of Tribonacci numbers given by T0 = 0, T1 = 1, T2 = 1, and Tn+3 = Tn+2 + Tn+1 + Tn for all n ≥ 0. The sequence of Tribonacci numbers is sequence A000073 on the OEIS. The first few terms of this sequence are given by (Tn )n≥0 = {0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, · · · }.
B 1
Mahadi Ddamulira [email protected]; [email protected] Institute of Analysis and Number Theory, University of Technology, Kopernikusgasse 24/II, 8010 Graz, Austria
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2 Main result In this paper, we study the problem of finding all Tribonacci numbers that are concatenations of two repdigits. More precisely, we completely solve the Diophantine equation q 2 10 − 1 10 − 1 2 Tn = d1 · · · d1 d2 · · · d2 = d1 · 10 + d2 , (2.1) 9 9 1 times
2 times
in non-negative integers (n, d1 , d2 , 1 , 2 ) with n ≥ 0, 1 ≥ 2 ≥ 1, and d1 , d2 ∈ {0, 1, . . . , 9}, d1 > 0. Our main result is the following. Theorem 2.1 The only Tribonacci numbers that are concatenations of two repdigits are Tn ∈ {13, 24, 44, 81}. Our method of proof involves the application of Baker’s theory for linear forms in logarithms of algebraic numbers, and the Baker-Davenport reduction procedure. Computations are done with the help of a computer program in Mathematica. 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, and (Bn )n≥0 be the sequence of balancing numbers given by B0 = 0, B1 = 1, and Bn+2 = 6Bn+1 − Bn for all n ≥ 0. This paper is inspired by the results of Alahmadi et al. [1], in which they show that the only Fibonacci numbers that are concatenations of two repdigits are Fn ∈ {13, 21, 34, 55, 89, 144, 233, 377}, and Rayaguru and Panda [10], who showed that Bn ∈ {35} is the only balancing number that can be written as a concatenation of two repdigits. Other related interesting results in this direction include: the result of Bravo and Luca [3],
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