scholarly journals Repdigits as Product of Fibonacci and Tribonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1720
Author(s):  
Dušan Bednařík ◽  
Eva Trojovská
Keyword(s):  
Lower Bounds ◽  
Continued Fractions ◽  
Decimal Expansion ◽  
Reduction Methods ◽  
Tribonacci Numbers

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö.

scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

scholarly journals Closed form expressions for two harmonic continued fractions

2017 ◽  
Vol 101 (552) ◽  
pp. 439-448 ◽  
Author(s):  
Martin Bunder ◽  
Joseph Tonien
Keyword(s):  
Closed Form ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Interested Reader ◽  
Continued Fraction Expansion ◽  
Decimal Expansion ◽  
Irrational Numbers ◽  
Image Position ◽  
Mathematical Constants

A continued fraction is an expression of the formThe expression can continue for ever, in which case it is called aninfinitecontinued fraction, or it can stop after some term, when we call it afinitecontinued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

Fibonacci numbers in generalized Pell sequences

2020 ◽  
Vol 70 (5) ◽  
pp. 1057-1068
Author(s):  
Jhon J. Bravo ◽  
Jose L. Herrera
Keyword(s):  
Lower Bounds ◽  
Continued Fractions ◽  
Fibonacci Numbers ◽  
Independent Interest ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Pell Numbers

AbstractIn this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

scholarly journals Repdigits as Product of Terms of k-Bonacci Sequences

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 682
Author(s):  
Petr Coufal ◽  
Pavel Trojovský
Keyword(s):  
Recent Work ◽  
Continued Fractions ◽  
Diophantine Equations ◽  
Reduction Method ◽  
Fibonacci Numbers ◽  
Decimal Expansion ◽  
Generalized Fibonacci Numbers ◽  
Initial Values ◽  
Transcendental Method

For any integer k≥2, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F−(k−2)(k)=⋯=F0(k)=0 and F1(k)=1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form aa…a, with a∈[1,9]) in the sequence (Fn(k)Fn(k+m))n, for m∈[1,9]. This result generalizes a recent work of Bednařík and Trojovská (the case in which (k,m)=(2,1)). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method).

scholarly journals Tribonacci numbers that are concatenations of two repdigits

2020 ◽  
Vol 114 (4) ◽  
Author(s):  
Mahadi Ddamulira
Keyword(s):  
Lower Bounds ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Reduction Procedure ◽  
Linear Forms ◽  
Tribonacci Numbers

Abstract Let $$ (T_{n})_{n\ge 0} $$ ( T n ) n ≥ 0 be the sequence of Tribonacci numbers defined by $$ T_0=0 $$ T 0 = 0 , $$ T_1=T_2=1$$ T 1 = T 2 = 1 , and $$ T_{n+3}= T_{n+2}+T_{n+1} +T_n$$ T n + 3 = T n + 2 + T n + 1 + T n for all $$ n\ge 0 $$ 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.

scholarly journals Irrationality measures for continued fractions with arithmetic functions

2015 ◽  
Vol 97 (111) ◽  
pp. 139-148
Author(s):  
Jaroslav Hancl ◽  
Kalle Leppälä
Keyword(s):  
Lower Bounds ◽  
Continued Fractions ◽  
Arithmetic Functions ◽  
Divisor Function ◽  
Euler Totient Function ◽  
Prime Factors

Let f(n) or the base-2 logarithm of f(n) be either d(n) (the divisor function), ?(n) (the divisor-sum function), ?(n) (the Euler totient function), ?(n) (the number of distinct prime factors of n) or ?(n) (the total number of prime factors of n). We present good lower bounds for |M/N ? ?| in terms of N, where ? = [0; f(1), f(2),...].

scholarly journals On Repdigits as Sums of Fibonacci and Tribonacci Numbers

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1774
Author(s):  
Pavel Trojovský
Keyword(s):  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Reduction Procedure ◽  
Linear Forms ◽  
Tribonacci Numbers ◽  
Symmetrical Type

In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a ”symmetrical” type of numbers) that can be written in the form Fn+Tn, for some n≥1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively.

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