One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation x(2) - 2(m) = y(n). Moreover, we prove that the diophantine equation x(2) - 2(m) = +/- y(n), x, y, m, n...
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One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation x(2) - 2(m) = y(n). Moreover, we prove that the diophantine equation x(2) - 2(m) = +/- y(n), x, y, m, n is an element of N, gcd(x, y) = 1, y > 1, n > 2 has only finitely many solutions, all of which satisfying n less than or equal to 7.310(5).
For k a parts per thousand yen 2, the k-generalized Fibonacci sequence (F (n) ((k)) ) (n) is defined by the initial values 0, 0, aEuro broken vertical bar, 0,1 (k terms) and such that each term afterwards is the sum o...
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For k a parts per thousand yen 2, the k-generalized Fibonacci sequence (F (n) ((k)) ) (n) is defined by the initial values 0, 0, aEuro broken vertical bar, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F (m) ((k)) = F (n) ((a"")) , with a"" > k > 1, n > a"" + 1, m > k + 1 are . In this paper, we confirm this conjecture.
In this paperwe prove that if {a, b, c} is a Diophantine triple with a < b < c, then{a+1, b, c} cannot be a Diophantine triple. Moreover, we show that if {a1, b, c} and {a(2), b, c} are Diophantine triples with ...
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In this paperwe prove that if {a, b, c} is a Diophantine triple with a < b < c, then{a+1, b, c} cannot be a Diophantine triple. Moreover, we show that if {a1, b, c} and {a(2), b, c} are Diophantine triples with a1 < a(2) < b < c < 16b(3), then{a(1), a(2), b, c} is a Diophantine quadruple. In view of these results, we conjecture that if {a1, b, c} and {a(2), b, c} are Diophantine triples with a(1) < a(2) < b < c, then {a(1), a(2), b, c} is a Diophantine quadruple.
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.
In this paper, we study the titular Diophantine equation for a fixed positive integer y >= 3 in nonnegative integers m, n, and a. We show that the nonnegative integer solutions (n, m, a) are finite in number, and w...
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In this paper, we study the titular Diophantine equation for a fixed positive integer y >= 3 in nonnegative integers m, n, and a. We show that the nonnegative integer solutions (n, m, a) are finite in number, and we provide a bound for them.
It is an open question of Baker whether the numbers L(1, chi) for nontrivial Dirichlet characters x with period q are linearly independent over Q. The best known result is due to Baker, Birch and Wirsing which affirms...
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It is an open question of Baker whether the numbers L(1, chi) for nontrivial Dirichlet characters x with period q are linearly independent over Q. The best known result is due to Baker, Birch and Wirsing which affirms this when q is co-prime to phi(q). In this paper, we extend their result. to any arbitrary family of moduli. More precisely, for a positive integer q, let X-q denote the set of all L(1, chi) values as chi varies over nontrivial Dirichlet characters with period q. Then for any finite set of pairwise co-prime natural numbers q(i), 1 <= i <= l with (q(1) ... q(l), phi(q(1)) ... phi(q(l))) = I, we show that the set X-q1 U ... U X-ql is linearly independent over Q. In the process, we also extend a result of Okada about linear independence of the cotangent values over Q as well as a result of Murty-Murty about (Q) over bar linear independence of such L(1, chi) values. Finally, we prove Q linear independence of such L values of Erdosian functions with distinct prime periods p(i) for 1 <= i <= l with (P-1 ... P-l, (P-1 ... P-l)) = 1.
Let P(m) denote the largest prime factor of an integer m >= 2, and put P(0) = P(1) = 1. For an integer k >= 2, let (F-n((k)))n >=(2-k) be the k-generalized Fibonacci sequence which starts with 0, ... , 0, 1 (...
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Let P(m) denote the largest prime factor of an integer m >= 2, and put P(0) = P(1) = 1. For an integer k >= 2, let (F-n((k)))n >=(2-k) be the k-generalized Fibonacci sequence which starts with 0, ... , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. Here, we show that if n >= k+2, then P(F-n((k))) > 0.01 root log n log log n. Furthermore, we determine all the k-Fibonacci numbers F-n((k)) whose largest prime factor is less than or equal to 7.
For a number field K, let zeta(K) (s) be the Dedekind zeta function associated to K. In this paper, we study non-vanishing and transcendence of zeta(K) as well as its derivative CK at s = 1/2. En route, we strengthen ...
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For a number field K, let zeta(K) (s) be the Dedekind zeta function associated to K. In this paper, we study non-vanishing and transcendence of zeta(K) as well as its derivative CK at s = 1/2. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of e(gamma) and non-vanishing of L-series at s = 1/2, J. Number Theory 161 (2016) 444 456].
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.
In this paper, we consider the D(-1)-triple {1, k(2) + 1, (k + 1)(2) + 1}. We extend the result obtained by Dujella. Filipin, and Fuchs (2007) [13] by determining the D(-1)-extension of this set. Moreover, we obtain a...
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In this paper, we consider the D(-1)-triple {1, k(2) + 1, (k + 1)(2) + 1}. We extend the result obtained by Dujella. Filipin, and Fuchs (2007) [13] by determining the D(-1)-extension of this set. Moreover, we obtain a D(1)-extension of the triple. (C) 2010 Elsevier Inc. All rights reserved.
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