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You searched for subject:(Primitive Divisors). Showing records 1 – 2 of 2 total matches.

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University of Colorado

1. Wakefield, Nathan Paul. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.

Degree: PhD, Mathematics, 2013, University of Colorado

Let (<em>gi</em>)<em>i</em> ≥1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (<em>fi</em>) <em>i</em> ≥1 by the following recursion: f1 = g1, <em>fn</em> = <em>gn</em> ∘ <em> fn</em>–1, for n ≥ 2. Choose α ∈ [special characters omitted] such that {[special characters omitted](α) : n ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if <em>f n</em>(α) = [special characters omitted] is written in lowest terms, then for all but finitely many n > 0, the numerator, <em>An</em>, has a primitive divisor; that is, there is a prime p which divides <em> An</em> but does not divide <em>Ai</em> for any i < n. In addition to the main result, several of the tools developed to prove the main result may be of interest. A key component of the main result was the development of a generalization of canonical height. Namely: If [f] is a set of rational maps, all commuting with a common function f, and f = [special characters omitted] is a generalized iteration of rational maps formed by <em>f n</em>(x) = <em>gn</em>(<em> fn</em>–1(x)) with <em> gi</em> coming from [f], then there is a unique canonical height funtion ĥf : K → [special characters omitted] which is identical to the canonical height function associated to f. Another key component of the main result was proving that under certain circumstances, being acted upon by a Chebyshev polynomial does not lead to significant differences between the size of the numerator and denominator of the result. Specifically, let γ ∈ {0, ±1, ±2} be fixed, and <em>gi</em> be a sequence of Chebyshev polynomials. Let f given by the following recurrence f 1(z) = g1(z), and <em>fi</em> = <em>gi</em>(<em> fi</em>–1(z)) for i ≥ 2. Pick any α ∈ [special characters omitted] with |α + γ| < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write <em>f n</em>(α + γ) − γ = [special characters omitted] in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed. Advisors/Committee Members: Su-Ion Ih, Katherine Stange, Robert Tubbs, Eric Stade, Juan Restrepo.

Subjects/Keywords: Arithmetic Dynamics; Chebyshev; Generalized Iteration; Primitive Divisors; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Wakefield, N. P. (2013). Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/25

Chicago Manual of Style (16th Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Doctoral Dissertation, University of Colorado. Accessed November 29, 2020. https://scholar.colorado.edu/math_gradetds/25.

MLA Handbook (7th Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Web. 29 Nov 2020.

Vancouver:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2020 Nov 29]. Available from: https://scholar.colorado.edu/math_gradetds/25.

Council of Science Editors:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/math_gradetds/25


Université de Bordeaux I

2. Dupuy, Benjamin. Etudes sur les équations de Ramanujan-Nagell et de Nagell-Ljunggren ou semblables : Impact of formulation and mixture of two pesticides (mesotrione and tebuconazole) on their biodegradation and microbial growth.

Degree: Docteur es, Mathématiques pures, 2009, Université de Bordeaux I

Dans cette thèse, on étudie deux types d’équations diophantiennes. Une première partie de notre étude porte sur la résolution des équations dites de Ramanujan-Nagell Cx2+ b2mD = yn. Une deuxième partie porte sur les équations dites de Ngell-Ljunggren xp+ypx+y = pezq incluant le cas diagonal p = q. Les nouveaux réesultats obtenus seront appliqués aux équations de la forme xp + yp = Bzq. L’équation de Catalan-Fermat (cas B = 1) fera l’objet d’un traitement à part.

In this thesis, we study two types of diophantine equations. A ?rst part of our study is about the resolution of the Ramanujan-Nagell equations Cx2 + b2mD = yn. A second part of our study is about the Nagell-Ljungren equations xp+yp x+y = pezq including the diagonal case p = q. Our new results will be applied to the diophantine equations of the form xp + yp = Bzq. The Fermat-Catalan equation (case B = 1) will be the subject of a special study.

Advisors/Committee Members: Bilu, Yuri (thesis director).

Subjects/Keywords: Nagell-Ljunggren; Ramanujan-Nagell; Formes linéaires en deux logarithmes; Nombres de Lucas; Nombres de Lehmer; Diviseurs primitifs; Théorie du corps de classe; Idéaux de Mihailescu généralisés; Nombres de classes; Idéal de Stickelberger; Entiers de Jacobi; Nagell-Ljunggren; Ramanujan-Nagell; Linear forms in two logarithms; Lucas numbers; Lehmers numbers; Primitive divisors; Class ?eld theory; Generalized Mihailescu ideals; Class number; Stickelberger ideal; Jacobi integers

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Dupuy, B. (2009). Etudes sur les équations de Ramanujan-Nagell et de Nagell-Ljunggren ou semblables : Impact of formulation and mixture of two pesticides (mesotrione and tebuconazole) on their biodegradation and microbial growth. (Doctoral Dissertation). Université de Bordeaux I. Retrieved from http://www.theses.fr/2009BOR13819

Chicago Manual of Style (16th Edition):

Dupuy, Benjamin. “Etudes sur les équations de Ramanujan-Nagell et de Nagell-Ljunggren ou semblables : Impact of formulation and mixture of two pesticides (mesotrione and tebuconazole) on their biodegradation and microbial growth.” 2009. Doctoral Dissertation, Université de Bordeaux I. Accessed November 29, 2020. http://www.theses.fr/2009BOR13819.

MLA Handbook (7th Edition):

Dupuy, Benjamin. “Etudes sur les équations de Ramanujan-Nagell et de Nagell-Ljunggren ou semblables : Impact of formulation and mixture of two pesticides (mesotrione and tebuconazole) on their biodegradation and microbial growth.” 2009. Web. 29 Nov 2020.

Vancouver:

Dupuy B. Etudes sur les équations de Ramanujan-Nagell et de Nagell-Ljunggren ou semblables : Impact of formulation and mixture of two pesticides (mesotrione and tebuconazole) on their biodegradation and microbial growth. [Internet] [Doctoral dissertation]. Université de Bordeaux I; 2009. [cited 2020 Nov 29]. Available from: http://www.theses.fr/2009BOR13819.

Council of Science Editors:

Dupuy B. Etudes sur les équations de Ramanujan-Nagell et de Nagell-Ljunggren ou semblables : Impact of formulation and mixture of two pesticides (mesotrione and tebuconazole) on their biodegradation and microbial growth. [Doctoral Dissertation]. Université de Bordeaux I; 2009. Available from: http://www.theses.fr/2009BOR13819

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