Displaying similar documents to “Common terms in binary recurrences”

Primefree shifted Lucas sequences

Lenny Jones (2015)

Acta Arithmetica

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We say a sequence = ( s ₙ ) n ≥ 0 is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, a = ( u ₙ ) n ≥ 0 , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences a ± k are simultaneously primefree. This...

On the intersection of two distinct k -generalized Fibonacci sequences

Diego Marques (2012)

Mathematica Bohemica

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Let k ≥ 2 and define F ( k ) : = ( F n ( k ) ) n ≥ 0 , the k -generalized Fibonacci sequence whose terms satisfy the recurrence relation F n ( k ) = F n - 1 ( k ) + F n - 2 ( k ) + ⋯ + F n - k ( k ) , with initial conditions 0 , 0 , ⋯ , 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 ) ∩ F ( m ) which gives...

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 square classes in generalized Fibonacci sequences

Zafer Şiar, Refik Keskin (2016)

Acta Arithmetica

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Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and U n + 1 = P U ₙ + Q U n - 1 , V n + 1 = P V ₙ + Q V n - 1 for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x². ...

Diophantine triples with values in binary recurrences

Clemens Fuchs, Florian Luca, Laszlo Szalay (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper, we study triples a , b and c of distinct positive integers such that a b + 1 , a c + 1 and b c + 1 are all three members of the same binary recurrence sequence.

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 ∞ 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...

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 ≥ 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 ≥ - ( 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...