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 and .
Bir Kafle, Florian Luca, Alain Togbé (2020)
Mathematica Bohemica
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We show that the only Lucas numbers which are factoriangular are and .
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 , in nonnegative integers , , , with .
Jhon J. Bravo, Jose L. Herrera (2020)
Archivum Mathematicum
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For an integer , let be the generalized Pell sequence which starts with ( terms) and each term afterwards is given by the linear recurrence . In this paper, we find all -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 . ...
Mariusz Skałba (2003)
Colloquium Mathematicae
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Consider a recurrence sequence of integers satisfying , where are fixed and a₀ ∈ -1,1. Assume that for all sufficiently large k. If there exists k₀∈ ℤ such that then for each negative integer -D there exist infinitely many rational primes q such that for some k ∈ ℕ and (-D/q) = -1.
Hayder R. Hashim (2022)
Archivum Mathematicum
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Let and be the Lucas sequences of the first and second kind respectively at the parameters and . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation where or with , . Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.
Curtis Cooper (2015)
Colloquium Mathematicae
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Melham discovered the Fibonacci identity . He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: . There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: . We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is .
Hayder Raheem Hashim, Szabolcs Tengely (2022)
Mathematica Bohemica
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Let be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are and , respectively. We show that the Diophantine equation has only finitely many solutions in , where , is even and . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral...
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 -Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the -Fibonacci numbers.
Refik Keskin, Zafer Şiar (2013)
Colloquium Mathematicae
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Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and for n ≥ 1, and V₀ = 2, V₁ = P and for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and . We show that there is no integer x such that when m ≥ 1 and r is an even integer. Also we completely solve the equation for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...
Shigeki Akiyama, Florian Luca (2013)
Acta Arithmetica
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We compare the growth of the least common multiple of the numbers and , where is a Lucas sequence and is some sequence of positive integers.
Zafer Şiar, Refik Keskin, Elif Segah Öztaş (2023)
Mathematica Bohemica
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Let and let be the -generalized Pell sequence defined by for with initial conditions In this study, we handle the equation in positive integers , , , such that and give an upper bound on Also, we will show that the equation with has only one solution given by
W. D. Gao, R. Thangadurai (2003)
Colloquium Mathematicae
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We study the structure of longest sequences in which have no zero-sum subsequence of length n (or less). We prove, among other results, that for and d arbitrary, or and d = 3, every sequence of c(n,d)(n-1) elements in which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where and .
Kouèssi Norbert Adédji, Japhet Odjoumani, Alain Togbé (2023)
Archivum Mathematicum
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Let and be the -th Padovan and Perrin numbers respectively. Let be non-zero integers with and , let be the generalized Lucas sequence given by , with and In this paper, we give effective bounds for the solutions of the following Diophantine equations where , and are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing 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 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 is -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.
Watcharapon Pimsert, Teerapat Srichan, Pinthira Tangsupphathawat (2023)
Czechoslovak Mathematical Journal
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We use the estimation of the number of integers such that belongs to an arithmetic progression to study the coprimality of integers in , , .
Fabien Durand (2011)
Journal of the European Mathematical Society
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The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let and be two multiplicatively independent Perron numbers. Then a sequence , where is a finite alphabet, is both -substitutive and -substitutive if and only if is ultimately...
Teerapat Srichan (2021)
Czechoslovak Mathematical Journal
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A natural number is said to be a -integer if , where and is not divisible by the th power of any prime. We study the distribution of such -integers in the Piatetski-Shapiro sequence with . As a corollary, we also obtain similar results for semi--free integers.
Gökhan Soydan, László Németh, László Szalay (2018)
Archivum Mathematicum
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Let denote the term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation in the positive integers and , where and are given positive integers. A complete solution is given if the exponents are included in the set . Based on the specific cases we could solve, and a computer search with we conjecture that beside the trivial solutions only , , and satisfy the title equation.
Reese Scott, Robert Styer (2013)
Journal de Théorie des Nombres de Bordeaux
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We consider , the number of solutions to the equation in nonnegative integers and integers , for given integers , , , and . When , we show that except for a finite number of cases all of which satisfy for each solution; when , we show that except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving solutions.