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Michel Zoeteman - One of the best experts on this subject based on the ideXlab platform.
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uniformly counting primes with a given primitive root and in an Arithmetic Progression
International Journal of Number Theory, 2019Co-Authors: Michel ZoetemanAbstract:We study the number of primes with a given primitive root and in an Arithmetic Progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, M...
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uniformly counting primes with a given primitive root and in an Arithmetic Progression
arXiv: Number Theory, 2018Co-Authors: Michel ZoetemanAbstract:We study the number of primes with a given primitive root and in an Arithmetic Progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
Enrique Gonzalezjimenez - One of the best experts on this subject based on the ideXlab platform.
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five squares in Arithmetic Progression over quadratic fields
Revista Matematica Iberoamericana, 2013Co-Authors: Enrique Gonzalezjimenez, Xavier XarlesAbstract:We give several criteria to show over which quadratic number fields Q(sqrt{D}) there should exists a non-constant Arithmetic Progressions of five squares. This is done by translating the problem to determining when some genus five curves C_D defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only non-constant Arithmetic Progressions of five squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2. Furthermore, we give an algorithm that allow to construct all the non-constant Arithmetic Progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.
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markoff rosenberger triples in Arithmetic Progression
Journal of Symbolic Computation, 2013Co-Authors: Enrique Gonzalezjimenez, Jose M TorneroAbstract:We study the solutions of the Rosenberg-Markoff equation ax^2+by^2+cz^2=dxyz (a generalization of the well-known Markoff equation). We specifically focus on looking for solutions in Arithmetic Progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x^2+y^2+z^2=dxyz over quadratic fields and the classic Markoff equation x^2+y^2+z^2=3xyz over an arbitrary number field.
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three cubes in Arithmetic Progression over quadratic fields
Archiv der Mathematik, 2010Co-Authors: Enrique GonzalezjimenezAbstract:We study the problem of the existence of Arithmetic Progressions of three cubes over quadratic number fields \({{\mathbb{Q}(\sqrt{D})}}\), where D is a squarefree integer. For this purpose, we give a characterization in terms of \({{\mathbb{Q}(\sqrt{D})}}\)-rational points on the elliptic curve E : y2 = x3 − 27. We compute the torsion subgroup of the Mordell–Weil group of this elliptic curve over \({{\mathbb{Q}(\sqrt{D})}}\) and we give an explicit answer, in terms of D, to the finiteness of the free part of \({E({\mathbb{Q}(\sqrt{D})})}\) for some cases. We translate this task to computing whether the rank of the quadratic D-twist of the modular curve X0(36) is zero or not.
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three cubes in Arithmetic Progression over quadratic fields
arXiv: Number Theory, 2009Co-Authors: Enrique GonzalezjimenezAbstract:We study the problem of the existence of Arithmetic Progressions of three cubes over quadratic number fields Q(sqrt(D)), where D is a squarefree integer. For this purpose, we give a characterization in terms of Q(sqrt(D))-rational points on the elliptic curve E:y^2=x^3-27. We compute the torsion subgroup of the Mordell-Weil group of this elliptic curve over Q(sqrt(D)) and we give partial answers to the finiteness of the free part of E(Q(sqrt(D))). This last task will be translated to compute if the rank of the quadratic D-twist of the modular curve X_0(36) is zero or not.
Xavier Xarles - One of the best experts on this subject based on the ideXlab platform.
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five squares in Arithmetic Progression over quadratic fields
Revista Matematica Iberoamericana, 2013Co-Authors: Enrique Gonzalezjimenez, Xavier XarlesAbstract:We give several criteria to show over which quadratic number fields Q(sqrt{D}) there should exists a non-constant Arithmetic Progressions of five squares. This is done by translating the problem to determining when some genus five curves C_D defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only non-constant Arithmetic Progressions of five squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2. Furthermore, we give an algorithm that allow to construct all the non-constant Arithmetic Progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.
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squares in Arithmetic Progression over number fields
Journal of Number Theory, 2012Co-Authors: Xavier XarlesAbstract:Abstract We show that there exists an upper bound for the number of squares in Arithmetic Progression over a number field that depends only on the degree of the field. We show that this bound is 5 for quadratic fields, and also that the result generalizes to k -powers for integers k > 1 .
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squares in Arithmetic Progression over number fields
arXiv: Algebraic Geometry, 2009Co-Authors: Xavier XarlesAbstract:We show that there exists an upper bound for the number of squares in Arithmetic Progression over a number field that depends only on the degree of the field. We show that this bound is 5 for quadratic fields, and also that the result generalizes to $k$-powers for $k>1$.
Alexandru Zaharescu - One of the best experts on this subject based on the ideXlab platform.
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partitions into k th powers of terms in an Arithmetic Progression
Mathematische Zeitschrift, 2018Co-Authors: Bruce C Berndt, Alexandru Zaharescu, Amita MalikAbstract:G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect kth powers, which was later proved by E. M. Wright. Recently, R. C. Vaughan provided a simpler asymptotic formula in the case $$k=2$$ . In this paper, we consider partitions into parts from a specific set $$A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb {N} , m \equiv a_0 \,(\text {mod}\,b_0) \right\} $$ , for fixed positive integers k, $$a_0,$$ and $$b_0$$ . We give an asymptotic formula for the number of such partitions, thus generalizing the results of Wright and Vaughan. Moreover, we prove that the number of such partitions is even (odd) infinitely often, which generalizes O. Kolberg’s theorem for the ordinary partition function p(n).
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Sums of Kloosterman sums over primes in an Arithmetic Progression
arXiv: Number Theory, 2018Co-Authors: Alexander Dunn, Alexandru ZaharescuAbstract:For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where $\text{Kl}_2(p;q)$ denotes a normalised Kloosterman sum with modulus $q$. This is a sparse analogue of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Mili\'cevi\'c showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use an optimisation argument inspired by Fouvry, Kowalski and Michel. Our argument compares three different bounds for bilinear forms involving Kloosterman sums. The first input in this method is a bilinear bound we prove using uniform asymptotics for oscillatory integrals due to Petrow, Kiral and Young. In contrast with the case when the sum runs over all primes, we exploit cancellation over a sum of stationary phase integrals that result from a Voronoi type summation. The second and third inputs are deep bilinear bounds for Kloosterman sums due to Fouvry-Kowalski-Michel and Kowalski-Michel-Sawin.
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on the farey fractions with denominators in Arithmetic Progression
Journal of Integer Sequences, 2006Co-Authors: Cristian Cobeli, Alexandru ZaharescuAbstract:Let F Q be the set of Farey fractions of order Q. Given the integers d ≥ 2 and 0 ≤ c ≤ d − 1, let F Q (c, d) be the subset of F Q of those fractions whose denominators are ≡ c (mod d), arranged in ascending order. The problem we address here is to show that as Q → ∞, there exists a limit probability measuring the distribution of s-tuples of consecutive denominators of fractions in F Q (c, d). This shows that the clusters of points (q0/Q, q1/Q, . . . , qs/Q) ∈ [0, 1] s+1 , where q0, q1, . . . , qs are consecutive denominators of members of F Q produce a limit set, denoted by D(c, d). The shape and the structure of this set are presented in several particular cases.
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On the Farey fractions with denominators in Arithmetic Progression
arXiv: Number Theory, 2005Co-Authors: Cristian Cobeli, Alexandru ZaharescuAbstract:Let $F_Q$ be the set of Farey fractions of order $Q$. Given the integers $\d\ge 2$ and $0\le \c \le \d-1$, let $F_Q(c,d)$ be the subset of $F_Q$ of those fractions whose denominators are $\equiv c \pmod d$, arranged in ascending order. The problem we address here is to show that as $Q\to\infty$, there exists a limit probability measuring the distribution of $s$-tuples of consecutive denominators of fractions in $F_Q(c,d)$. This shows that the clusters of points $(q_0/Q,q_1/Q,...,q_s/Q)\in[0,1]^{s+1}$, where $q_0,q_1,...,q_s$ are consecutive denominators of members of $F_Q$ produce a limit set, denoted by $D(c,d)$. The shape and the structure of this set are presented in several particular cases.
Carlo Sanna - One of the best experts on this subject based on the ideXlab platform.
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Covering an Arithmetic Progression with geometric Progressions and vice versa
International Journal of Number Theory, 2014Co-Authors: Carlo SannaAbstract:We show that there exists a positive constant C such that the following holds: Given an infinite Arithmetic Progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric Progressions to cover the first n terms of A. A similar result is presented, with the role of Arithmetic and geometric Progressions reversed.