Can a Lucas number be a sum of three repdigits?

Chèfiath A. Adegbindin; Alain Togbé

Displaying similar documents to “Can a Lucas number be a sum of three repdigits?”

Zero-sums of length kq in $ℤ_{q}^{d}$

Silke Kubertin (2005)

Acta Arithmetica

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On sums of powers of the positive integers

A. Schinzel (2013)

Colloquium Mathematicae

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The pairs (k,m) are studied such that for every positive integer n we have $1^{k} + 2^{k} + \dots + n^{k} | 1^{k m} + 2^{k m} + \dots + n^{k m}$ .

An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers

Carlos Alexis Gómez Ruiz, Florian Luca (2014)

Colloquium Mathematicae

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A generalization of the well-known Fibonacci sequence ${F ₙ}_{n \geq 0}$ given by F₀ = 0, F₁ = 1 and $F_{n + 2} = F_{n + 1} + F ₙ$ for all n ≥ 0 is the k-generalized Fibonacci sequence ${F ₙ^{(k)}}_{n \geq - (k - 2)}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $F ₙ ² + F ²_{n + 1} ² = F_{2 n + 1}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This...

On the distance between generalized Fibonacci numbers

Jhon J. Bravo, Carlos A. Gómez, Florian Luca (2015)

Colloquium Mathematicae

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For an integer k ≥ 2, let $(F ₙ^{(k)}) ₙ$ be the k-Fibonacci sequence which starts with 0,..., 0,1 (k terms) and each term afterwards is the sum of the k preceding terms. This paper completes a previous work of Marques (2014) which investigated the spacing between terms of distinct k-Fibonacci sequences.

The sum of digits of $⌊ n^{c} ⌋$

Johannes F. Morgenbesser (2011)

Acta Arithmetica

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On the spacing between terms of generalized Fibonacci sequences

Diego Marques (2014)

Colloquium Mathematicae

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For k ≥ 2, the k-generalized Fibonacci sequence $(F ₙ^{(k)}) ₙ$ is defined to have the initial k terms 0,0,...,0,1 and be such that each term afterwards is the sum of the k preceding terms. We will prove that the number of solutions of the Diophantine equation $F ₘ^{(k)} - F ₙ^{(ℓ)} = c > 0$ (under some weak assumptions) is bounded by an effectively computable constant depending only on c.

On the intersection of two distinct $k$ -generalized Fibonacci sequences

Diego Marques (2012)

Mathematica Bohemica

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Let $k \geq 2$ and define $F^{(k)} : = {(F_{n}^{(k)})}_{n \geq 0}$ , the $k$ -generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_{n}^{(k)} = F_{n - 1}^{(k)} + F_{n - 2}^{(k)} + \dots + F_{n - k}^{(k)}$ , with initial conditions $0, 0, \dots, 0, 1$ ( $k$ terms) and such that the first nonzero term is $F_{1}^{(k)} = 1$ . The sequences $F : = F^{(2)}$ and $T : = F^{(3)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_{n}^{(k)} = F_{m}^{(ℓ)}$ . In this note, we use transcendental tools to provide a general method for finding the intersections $F^{(k)} \cap F^{(m)}$ which gives...

A note on signs of Kloosterman sums

Kaisa Matomäki (2011)

Bulletin de la Société Mathématique de France

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We prove that the sign of Kloosterman sums $Kl (1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.

Universal forms $\sum a_{i} {(x_{i})}^{n}$ and Waring’s problem

L. Dickson (1936)

Acta Arithmetica

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On the 4-norm of an automorphic form

Valentin Blomer (2013)

Journal of the European Mathematical Society

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We prove the optimal upper bound $\sum_{f} {∥f∥}_{4}^{4} ≪ q^{ϵ}$ where $f$ runs over an orthonormal basis of Maass cusp forms of prime level $q$ and bounded spectral parameter.

On the $q$ -Pell sequences and sums of tails

Alexander E. Patkowski (2017)

Czechoslovak Mathematical Journal

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We examine the $q$ -Pell sequences and their applications to weighted partition theorems and values of $L$ -functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous...

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

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Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_{N}^{(m)} (ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_{N}^{(m)} (ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

Some interpretations of the $(k, p)$ -Fibonacci numbers

Natalia Paja, Iwona Włoch (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the $(k, p)$ -Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the $(k, p)$ -Fibonacci numbers.

On some universal sums of generalized polygonal numbers

Fan Ge, Zhi-Wei Sun (2016)

Colloquium Mathematicae

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For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as $p ₃ (x) + p ₅ (y) + p_{11} (z)$ and 3p₃(x) + p₅(y) + p₇(z) with x,y,z...

Fermat $k$ -Fibonacci and $k$ -Lucas numbers

Jhon J. Bravo, Jose L. Herrera (2020)

Mathematica Bohemica

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Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all $k$ -Fibonacci and $k$ -Lucas numbers which are Fermat numbers. Some more general results are given.

Exponential sums of the form $\sum χ {(x)}^{a x} ζ_{m}^{b x}$

S. Gurak (2007)

Acta Arithmetica

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On Sums of Squares in $Q^{\frac{1}{2}} (X)$ etc

W. J. Ellison (1970-1971)

Séminaire de théorie des nombres de Bordeaux

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On the golden number and Fibonacci type sequences

Eugeniusz Barcz (2019)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The paper presents, among others, the golden number $ϕ$ as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein’s theorem. To demonstrate the equality , where $f_{n}$ is $n$ -th Fibonacci number also the formula from Corollary has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified.

Multiplicatively dependent triples of Tribonacci numbers

Carlos Alexis Ruiz Gómez, Florian Luca (2015)

Acta Arithmetica

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We consider the Tribonacci sequence $T : = {T_{n}}_{n \geq 0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n + 3} = T_{n + 2} + T_{n + 1} + T_{n}$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

A new proof of the $q$ -Dixon identity

Victor J. W. Guo (2018)

Czechoslovak Mathematical Journal

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We give a new and elementary proof of Jackson’s terminating $q$ -analogue of Dixon’s identity by using recurrences and induction.