We find all positive integer solutions of the Diophantine equation F (n) + F (m) + F (e) = 2 (a) , where F (k) is the kth term of the Fibonacci sequence. This paper continues and extends the previous work of J.J. Brav...
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We find all positive integer solutions of the Diophantine equation F (n) + F (m) + F (e) = 2 (a) , where F (k) is the kth term of the Fibonacci sequence. This paper continues and extends the previous work of J.J. Bravo and F. Luca [On the Diophantine equation F (n) + F (m) = 2 (a) , Quaest. Math., to appear].
Let (F-n) n >= 0 be the Fibonacci sequence given by Fm+2 = Fm+1 + F-m, for m >= 0, where F-0 = 0 and F-1 = 1. In 2011, Luca and Oyono proved that if F-m(s) + F-m+1(s) is a Fibonacci number, with m >= 2, then ...
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Let (F-n) n >= 0 be the Fibonacci sequence given by Fm+2 = Fm+1 + F-m, for m >= 0, where F-0 = 0 and F-1 = 1. In 2011, Luca and Oyono proved that if F-m(s) + F-m+1(s) is a Fibonacci number, with m >= 2, then s = 1 or 2. A well-known generalization of the Fibonacci sequence, is the k-generalized Fibonacci sequence (F-n((k)))(n) which is defined by the initial values 0,0,..., 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we generalize Luca and Oyono's method by proving that the Diophantine equation (F-m((k)))(s) + (F-m+1((k)))(s) = F-n((k)) has no solution in positive integers n,m,k and s, if 3 <= k <= min{m, log s}. (C) 2015 Published by Elsevier Inc.
We prove that for each odd number k, the sequence (k2(n) + 1)(n >= 1) contains only a finite number of Carmichael numbers. We also prove that k = 27 is the smallest value for which such a sequence contains some Car...
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We prove that for each odd number k, the sequence (k2(n) + 1)(n >= 1) contains only a finite number of Carmichael numbers. We also prove that k = 27 is the smallest value for which such a sequence contains some Carmichael number.
For an integer , let be the -Fibonacci sequence which starts with ( terms) and each term afterwards is the sum of the preceding terms. In this paper, we find all repdigits (i.e., numbers with only one distinct digit i...
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For an integer , let be the -Fibonacci sequence which starts with ( terms) and each term afterwards is the sum of the preceding terms. In this paper, we find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion) which are sums of two -Fibonacci numbers. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker-Davenport reduction method. This paper is an extended work related to our previous work (Bravo and Luca Publ Math Debr 82:623-639, 2013).
For any periodic function f : N -> C with period g, we study the Dirichlet series L(s, f) := Sigma(n >= 1) f(n)/n(s). It is well-known that this admits an analytic continuation to the entire complex plane except...
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For any periodic function f : N -> C with period g, we study the Dirichlet series L(s, f) := Sigma(n >= 1) f(n)/n(s). It is well-known that this admits an analytic continuation to the entire complex plane except at s = 1, where it has a simple pole with residue rho := q(-1) Sigma(1 <= a <= q) f(a). Thus, the function is analytic at s = 1 when rho = 0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L(1, f) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L(1, f). (C) 2014 Elsevier Inc. All rights reserved.
Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. For a wide class of sets A, we prove an analogous result for the function pA(n) that counts partitions of ...
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Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. For a wide class of sets A, we prove an analogous result for the function pA(n) that counts partitions of n into terms belonging to A.
For k >= 2, the k-generalized Fibonacci sequence (F-n((k)))(n) is defined by the initial values 0, 0, ..., 0, 1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we fi...
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For k >= 2, the k-generalized Fibonacci sequence (F-n((k)))(n) is defined by the initial values 0, 0, ..., 0, 1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we find all generalized Fibonacci numbers written in the form 2(a) + 3(b) + 5(c). This work generalizes a recent Marques-Togbe result [20] concerning the case k = 2.
In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an S-unit whose coefficients are bounded. In particular, we fin...
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In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an S-unit whose coefficients are bounded. In particular, we find the largest member of the Fibonacci sequence which can be written as a sum or a difference between a factorial and an S-unit associated to the set of primes {2, 3, 5, 7}.
A positive integer n is said to be a perfect number, if sigma-(n) = 2n, where sigma (N) is the sum of all positive divisors of N. Luca (Rend Circ Mat Palermo Ser 49:313-318, 2000) proved that there is no perfect numbe...
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A positive integer n is said to be a perfect number, if sigma-(n) = 2n, where sigma (N) is the sum of all positive divisors of N. Luca (Rend Circ Mat Palermo Ser 49:313-318, 2000) proved that there is no perfect number in the Fibonacci sequence. For k >= 2, the k-generalized Fibonacci sequence (F-n((k)))(n) is defined by the initial values 0, 0, ... 0, 1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we prove, among other things, that there is no even perfect numbers belonging to k-generalized Fibonacci sequences when k not equivalent to 3 (mod 4).
Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant gamma. The proof is by reduction to known irrationality criteria for gamma involvin...
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Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant gamma. The proof is by reduction to known irrationality criteria for gamma involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linearforms in 1, gamma, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.
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