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

Jhon J. Bravo; Jose L. Herrera

Displaying similar documents to “Fermat $k$ -Fibonacci and $k$ -Lucas numbers”

Pell and Pell-Lucas numbers of the form $- 2^{a} - 3^{b} + 5^{c}$

Yunyun Qu, Jiwen Zeng (2020)

Czechoslovak Mathematical Journal

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In this paper, we find all Pell and Pell-Lucas numbers written in the form $- 2^{a} - 3^{b} + 5^{c}$ , in nonnegative integers $a$ , $b$ , $c$ , with $0 \leq max {a, b} \leq c$ .

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

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We show that the only Lucas numbers which are factoriangular are $1$ and $2$ .

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 a generalization of the Pell sequence

Jhon J. Bravo, Jose L. Herrera, Florian Luca (2021)

Mathematica Bohemica

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The Pell sequence ${(P_{n})}_{n = 0}^{\infty}$ is the second order linear recurrence defined by $P_{n} = 2 P_{n - 1} + P_{n - 2}$ with initial conditions $P_{0} = 0$ and $P_{1} = 1$ . In this paper, we investigate a generalization of the Pell sequence called the $k$ -generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers...

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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Repdigits in generalized Pell sequences

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

Archivum Mathematicum

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For an integer $k \geq 2$ , let ${(n)}_{n}$ be the $k -$ generalized Pell sequence which starts with $0, ..., 0, 1$ ( $k$ terms) and each term afterwards is given by the linear recurrence $n = 2 n - 1 + n - 2 + \dots + n - k$ . In this paper, we find all $k$ -generalized Pell numbers with only one distinct digit (the so-called repdigits). 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 repdigits in the usual Pell sequence ${(P_{n}^{(2)})}_{n}$ . ...

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...

Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $F_{n}$ be the $n$ -th Fibonacci number. Put $ϕ = \frac{1 + \sqrt{5}}{2}$ . We prove that the following inequalities hold for any real $α$ : 1) ${inf}_{n \in ℕ} | | F_{n} α | | \leq \frac{ϕ - 1}{ϕ + 2}$ , 2) ${lim inf}_{n \to \infty} | | F_{n} α | | \leq \frac{1}{5}$ , 3) ${lim inf}_{n \to \infty} | | ϕ^{n} α | | \leq \frac{1}{5}$ . These results are the best possible.

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.

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.

On a fixobject property for $l^{c} β$ -semigroupoids

M. Đurić (1973)

Matematički Vesnik

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An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

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On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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On perfect powers in $k$ -generalized Pell sequence

Zafer Şiar, Refik Keskin, Elif Segah Öztaş (2023)

Mathematica Bohemica

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Let $k \geq 2$ and let ${(P_{n}^{(k)})}_{n \geq 2 - k}$ be the $k$ -generalized Pell sequence defined by $P_{n}^{(k)} = 2 P_{n - 1}^{(k)} + P_{n - 2}^{(k)} + \dots + P_{n - k}^{(k)}$ for $n \geq 2$ with initial conditions $P_{- (k - 2)}^{(k)} = P_{- (k - 3)}^{(k)} = \dots = P_{- 1}^{(k)} = P_{0}^{(k)} = 0, P_{1}^{(k)} = 1 .$ In this study, we handle the equation $P_{n}^{(k)} = y^{m}$ in positive integers $n$ , $m$ , $y$ , $k$ such that $k, y \geq 2,$ and give an upper bound on $n .$ Also, we will show that the equation $P_{n}^{(k)} = y^{m}$ with $2 \leq y \leq 1000$ has only one solution given by $P_{7}^{(2)} = 13^{2} .$

The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

Matematički Vesnik

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The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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Displaying similar documents to “Fermat k -Fibonacci and k -Lucas numbers”

Displaying similar documents to “Fermat $k$ -Fibonacci and $k$ -Lucas numbers”