Tesi sul tema "Diophantine equations"
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1
Lee, Ellyn Jay. "Studies on diophantine equations". Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239075.
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Mabaso, Automan Sibusiso. "Some exponential diophantine equations". Thesis, Stellenbosch : Stellenbosch University, 2013. http://hdl.handle.net/10019.1/85587.
Testo completoAbstract (sommario):
Thesis (MSc)--Stellenbosch University, 2013.ENGLISH ABSTRACT: The aim of this thesis is to study some methods used in solving exponential Diophan- tine equations. There is no generic method or algorithm that can be used in solving all Diophantine equations. The main focus for our study will be solving the exponential Dio- phantine equations using the modular approach and the linear forms in two logarithms approach.
AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om sommige metodes te bestudeer om sekere Diophantiese vergelykings op te los. Daar is geen metode wat alle Diophantiese vergelykings kan oplos nie. Die fokus van os studie is hoofsaaklik om eksponensiele Diophantiese vergelykings op te los met die modul^ere metode en met die metode van line^ere vorms in twee logaritmes.
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Bartolomé, Boris. "Diophantine equations and cyclotomic fields". Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0104/document.
Testo completoAbstract (sommario):
Cette thèse examine quelques approches aux équations diophantiennes, en particulier les connexions entre l’analyse diophantienne et la théorie des corps cyclotomiques.Tout d’abord, nous proposons une introduction très sommaire et rapide aux méthodes d’analyse diophantienne que nous avons utilisées dans notre travail de recherche. Nous rappelons la notion de hauteur et présentons le PGCD logarithmique.Ensuite, nous attaquons une conjecture, formulée par Skolem en 1937, sur une équation diophantienne exponentielle. Pour cette conjecture, soit K un corps de nombres, α1 ,…, αm , λ1 ,…, λm des éléments non-nuls de K, et S un ensemble fini de places de K (qui contient toutes les places infinies), de telle sorte que l’anneau de S-entiers OS = OK,S = {α ∈ K : |α|v ≤ 1 pour les places v ∈/ S}contienne α1 , . . . , αm , λ1 , . . . , λm α1-1 , . . . , αm-1. Pour chaque n ∈ Z, soit A(n)=λ_1 α_1^n+⋯+λ_m α_m^n∈O_S. Skolem a suggéré [SK1] :Conjecture (principe local-global exponentiel). Supposons que pour chaque idéal non-nul a de l’anneau O_S, il existe n ∈ Z tel que A(n) ≡0 mod a. Alors, il existe n ∈ Z tel que A(n)=0.Soit Γ le groupe multiplicatif engendré par α1 ,…, αm. Alors Γ est le produit d’un groupe abélien fini et d’un groupe libre de rang fini. Nous démontrons que cette conjecture est vraie lorsque le rang de Γ est un.Après cela, nous généralisons un résultat précédent de Mourad Abouzaid ([A]). Soit F (X,Y) ∈ Q[X,Y] un Q-polynôme irréductible. En 2008, Mourad Abouzaid [A] a démontré le théorème suivant:Théorème (Abouzaid). Supposons que (0,0) soit un point non-singulier de la courbe plane F(X,Y) = 0. Soit m = degX F, n = degY F, M = max{m, n}. Soit ε tel que 0 < ε < 1. Alors, pour toute solution (α, β) ∈ Q ̅2 de F(X,Y) = 0, nous avons soit max{h(α), h(β)} ≤ 56M8ε−2hp(F) + 420M10ε−2 log(4M),soitmax{|h(α) − nlgcd(α, β)|,|h(β) − mlgcd(α, β)|} ≤ εmax{h(α), h(β)}++ 742M7ε−1hp(F) + 5762M9ε−1log(2m + 2n)Cependant, il a imposé la condition que (0,0) soit un point non-singulier de la courbe plane F(X,Y) = 0. En utilisant des versions quelque peu différentes du lemme “absolu” de Siegel et du lemme d’Eisenstein, nous avons pu lever la condition et démontrer le théorème de façon générale. Nous démontrons le théorème suivant:Théorème. Soit F(X,Y) ∈ Q ̅[X,Y] un polynôme absolument irréductible qui satisfasse F(0,0)=0. Soit m=degX F, n=degY F et r = min{i+j:(∂^(i+j) F)/(∂^i X∂^j Y)(0,0)≠0}. Soit ε tel que 0 < ε < 1. Alors, pour tout (α, β) ∈ Q ̅2 tel que F(α,β) = 0, nous avons soith(α) ≤ 200ε−2mn6(hp(F) + 5)soit|(lgcd(α,β))/r-h(α)/n|≤1/r (εh(α)+4000ε^(-1) n^4 (h_p (F)+log(mn)+1)+30n^2 m(h_p (F)+log(mn) ))Ensuite, nous donnons un aperçu des outils que nous avons utilisés dans les corps cyclotomiques. Nous tentons de développer une approche systématique pour un certain genre d’équations diophantiennes. Nous proposons quelques résultats sur les corps cyclotomiques, les anneaux de groupe et les sommes de Jacobi, qui nous seront utiles pour ensuite décrire l’approche.Finalement, nous développons une application de l’approche précédemment expliquée. Nous considèrerons l’équation diophantienne(1) Xn − 1 = BZn,où B ∈ Z est un paramètre. Définissons ϕ∗(B) := ϕ(rad (B)), où rad (B) est le radical de B, et supposons que(2) (n, ϕ∗(B)) = 1.Pour B ∈ N_(>1) fixé, soit N(B) = {n ∈ N_(>1) | ∃ k > 0 tel que n|ϕ∗(B)}. Si p est un premier impair, nous appellerons CF les conditions combinéesI La conjecture de Vandiver est vraie pour p, c’est-à-dire que le nombre de classe h+ du sous-corps réel maximal du corps cyclotomique Q[ζp ], n’est pas divisible par p.II Nous avons ir(p) < √p − 1, en d’autre mots, il y a au plus √p − 1 entiers impairs k < p tels que le nombre de Bernouilli Bk ≡ 0 mod p. [...]This thesis examines some approaches to address Diophantine equations, specifically we focus on the connection between the Diophantine analysis and the theory of cyclotomic fields.First, we propose a quick introduction to the methods of Diophantine approximation we have used in this research work. We remind the notion of height and introduce the logarithmic gcd.Then, we address a conjecture, made by Thoralf Skolem in 1937, on an exponential Diophantine equation. For this conjecture, let K be a number field, α1 ,…, αm , λ1 ,…, λm non-zero elements in K, and S a finite set of places of K (containing all the infinite places) such that the ring of S-integersOS = OK,S = {α ∈ K : |α|v ≤ 1 pour les places v ∈/ S}contains α1 , . . . , αm , λ1 , . . . , λm α1-1 , . . . , αm-1. For each n ∈ Z, let A(n)=λ_1 α_1^n+⋯+λ_m α_m^n∈O_S. Skolem suggested [SK1] :Conjecture (exponential local-global principle). Assume that for every non zero ideal a of the ring O_S, there exists n ∈ Z such that A(n) ≡0 mod a. Then, there exists n ∈ Z such that A(n)=0.Let Γ be the multiplicative group generated by α1 ,…, αm. Then Γ is the product of a finite abelian group and a free abelian group of finite rank. We prove that the conjecture is true when the rank of Γ is one.After that, we generalize a result previously published by Abouzaid ([A]). Let F(X,Y) ∈ Q[X,Y] be an irreducible Q-polynomial. In 2008, Abouzaid [A] proved the following theorem:Theorem (Abouzaid). Assume that (0,0) is a non-singular point of the plane curve F(X,Y) = 0. Let m = degX F, n = degY F, M = max{m, n}. Let ε satisfy 0 < ε < 1. Then for any solution (α,β) ∈ Q ̅2 of F(X,Y) = 0, we have eithermax{h(α), h(β)} ≤ 56M8ε−2hp(F) + 420M10ε−2 log(4M),ormax{|h(α) − nlgcd(α, β)|,|h(β) − mlgcd(α, β)|} ≤ εmax{h(α), h(β)}++ 742M7ε−1hp(F) + 5762M9ε−1log(2m + 2n)However, he imposed the condition that (0, 0) be a non-singular point of the plane curve F(X,Y) = 0. Using a somewhat different version of Siegel’s “absolute” lemma and of Eisenstein’s lemma, we could remove the condition and prove it in full generality. We prove the following theorem:Theorem. Let F(X,Y) ∈ Q ̅[X,Y] be an absolutely irreducible polynomial satisfying F(0,0)=0. Let m=degX F, n=degY F and r = min{i+j:(∂^(i+j) F)/(∂^i X∂^j Y)(0,0)≠0}. Let ε be such that 0 < ε < 1. Then, for all (α, β) ∈ Q ̅2 such that F(α,β) = 0, we have eitherh(α) ≤ 200ε−2mn6(hp(F) + 5)or|(lgcd(α,β))/r-h(α)/n|≤1/r (εh(α)+4000ε^(-1) n^4 (h_p (F)+log(mn)+1)+30n^2 m(h_p (F)+log(mn) ))Then, we give an overview of the tools we have used in cyclotomic fields. We try there to develop a systematic approach to address a certain type of Diophantine equations. We discuss on cyclotomic extensions and give some basic but useful properties, on group-ring properties and on Jacobi sums.Finally, we show a very interesting application of the approach developed in the previous chapter. There, we consider the Diophantine equation(1) Xn − 1 = BZn,where B ∈ Z is understood as a parameter. Define ϕ∗(B) := ϕ(rad (B)), where rad (B) is the radical of B, and assume that (2) (n, ϕ∗(B)) = 1.For a fixed B ∈ N_(>1)we let N(B) = {n ∈ N_(>1) | ∃ k > 0 such that n|ϕ∗(B)}. If p is an odd prime, we shall denote by CF the combined condition requiring thatI The Vandiver Conjecture holds for p, so the class number h+ of the maximal real subfield of the cyclotomic field Q[ζp ] is not divisible by p.II We have ir>(p) < √p − 1, in other words, there is at most √p − 1 odd integers k < p such that the Bernoulli number Bk ≡ 0 mod p. [...]
4
Yesilyurt, Deniz. "Solving Linear Diophantine Equations And Linear Congruential Equations". Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-19247.
Testo completoAbstract (sommario):
This report represents GCD, euclidean algorithm, linear diophantine equation and linear congruential equation. It investigates the methods for solving linear diophantine equations and linear congruential equations in several variables. There are many examples which illustrate the methods for solving equations.
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Smart, Nigel Paul. "The computer solution of diophantine equations". Thesis, University of Kent, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.315089.
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Long, Rachel Louise. "The algorithmic solution of simultaneous diophantine equations". Thesis, Oxford Brookes University, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444340.
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Baczkowski, Daniel M. "Diophantine Equations Involving Arithmetic Functions of Factorials". Miami University / OhioLINK, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=miami1088086258.
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Akhtari, Shabnam. "Thue equations and related topics". Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/1288.
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Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive
integers X and Y. Since there are infinitely many pairs (a, b) for which two
such solutions exist, this result is sharp. It is also effectively proved that
for fixed positive integers a and b, there are at most two positive integer
solutions to the quartic Diophantine equation
aX⁴ - bY² = 2.
We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas.
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Schindler, Damaris. "On diophantine equations involving norm forms and bihomogeneous forms". Thesis, University of Bristol, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.601162.
Testo completoAbstract (sommario):
If the descent theory, developed by Colliot-Thelene and Sansuc, applies, then it can reduce the question of understanding whether the Brauer-Manin obstruction is the only one to understanding weak approximation on the resulting descent varieties. In some cases the descent varieties are easier to handle and accessible by analytic methods as for example t he circle method. In joint work with A. Skorobogatov we followed this approach focusing on varieties corresponding to the representation of a norm form by a product of linear polynomials. We present this work in the first part of this thesis which involves an application of the circle method over number fields to systems of linear equations involving norm forms. In the second part of this thesis we study the arithmetic of subvarieties in biprojective space. So far, the circle method has been a very useful tool to prove many cases of Manin's conjecture. Work of B. Birch back in 1962 establishes this for smooth complete intersections in projective space as soon as the number of variables is large enough depending on the degree and number of equations. In biprojective space there is not much known so far, unless the underlying polynomials are of bidegree (1,1). A combination of the circle method with the generalised hyperbola method recently developed by V. Blomer and J. Brudern allows us to verify Manin 's conjecture for certain smooth hypersurfaces of general bidegree in biprojective space.
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Ibrahim, Mostafa. "Modular and reciprocity approaches to a family of diophantine equations". Thesis, University of Warwick, 2009. http://wrap.warwick.ac.uk/2761/.
Testo completoAbstract (sommario):
In this thesis we study the Diophantine equation xp - Dy2p = z2; gcd(x; z) = 1; p prime: We combine two approaches: - The modular approach using in Wiles's proof of Fermat's Last Theorem. - Elementary quadratic reciprocity. We show how using this combination of approaches and computer calculations we can get congruence conditions for the exponent p.
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蔡國光 e Kwok-kwong Stephen Choi. "Some explicit estimates on linear diophantine equations in three primevariables". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1990. http://hub.hku.hk/bib/B3120966X.
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Kaminski, Lance. "A discussion of homogenous quadratic equations". Thesis, Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/1630.
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Smith, Jason. "Solvability characterizations of Pell like equations". [Boise, Idaho] : Boise State University, 2009. http://scholarworks.boisestate.edu/td/55/.
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Ma, Fei Chun. "A Diophantine equations based cipher for Internet EDI security in Macau". Thesis, University of Macau, 1997. http://umaclib3.umac.mo/record=b1445591.
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Choi, Kwok-kwong Stephen. "Some explicit estimates on linear diophantine equations in three prime variables /". [Hong Kong] : University of Hong Kong, 1990. http://sunzi.lib.hku.hk/hkuto/record.jsp?B12907236.
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Ren, Ai. "Embedded Surface Attack on Multivariate Public Key Cryptosystems from Diophantine Equation". University of Cincinnati / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1558364211159262.
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Beauchamp, Bradley K. McCrone Sharon Rich Beverly Susan. "Exploring calculus students' understanding of L'Hôpital's Rule". Normal, Ill. : Illinois State University, 2006. http://proquest.umi.com/pqdweb?index=0&did=1273094441&SrchMode=1&sid=3&Fmt=2&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1181240966&clientId=43838.
Testo completoAbstract (sommario):
Thesis (Ph. D.)--Illinois State University, 2006.Title from title page screen, viewed on June 7, 2007. Dissertation Committee: Dissertation Committee: Sharon S. McCrone, Beverly S. Rich (co-chairs), James F. Cottrill, Lucian L. Ionescu. Includes bibliographical references (leaves 155-159) and abstract. Also available in print.
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Allen, Patrick. "Multiplicities of Linear Recurrence Sequences". Thesis, University of Waterloo, 2006. http://hdl.handle.net/10012/2942.
Testo completoAbstract (sommario):
In this report we give an overview of some of the major results concerning the multiplicities of linear recurrence sequences. We first investigate binary recurrence sequences where we exhibit a result due to Beukers and a result due to Brindza, Pintér and Schmidt. We then investigate ternary recurrences and exhibit a result due to Beukers building on work of Beukers and Tijdeman. The last two chapters deal with a very important result due to Schmidt in which we bound the zero-multiplicity of a linear recurrence sequence of order t by a function involving t alone. Moreover we improve on Schmidt's bound by making some minor changes to his argument.
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Heimonen, A. (Ari). "On effective irrationality measures for some values of certain hypergeometric functions". Doctoral thesis, University of Oulu, 1997. http://urn.fi/urn:isbn:9514247191.
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Abstract
The dissertation consists of three articles in which irrationality measures for some values of certain special cases of the Gauss hypergeometric function are considered in both archimedean and non-archimedean metrics.
The first presents a general result and a divisibility criterion for certain products of binomial coefficients upon which the sharpenings of the general result in special cases rely. The paper also provides an improvement concerning th e values of the logarithmic function. The second paper includes two other special cases, the first of which gives irrationality measures for some values of the arctan function, for example, and the second concerns values of the binomial function. All the results of the first two papers are effective, but no computation of the constants for explicit presentation is carried out. This task is fulfilled in the third article for logarithmic and binomial cases. The results of the latter case are applied to some Diophantine equations.
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Dumke, Jan Henrik Verfasser], Jörg [Akademischer Betreuer] [Brüdern e Valentin [Akademischer Betreuer] Blomer. "Diophantine Equations in Many Variables / Jan Henrik Dumke. Gutachter: Jörg Brüdern ; Valentin Blomer. Betreuer: Jörg Brüdern". Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2014. http://d-nb.info/1060543192/34.
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Wodzak, Michael A. "Entire functions and uniform distribution /". free to MU campus, to others for purchase, 1996. http://wwwlib.umi.com/cr/mo/fullcit?p9823328.
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Bartolomé, Boris Verfasser], Preda [Akademischer Betreuer] Mihăilescu, Yuri [Akademischer Betreuer] Bilu, Yann [Akademischer Betreuer] Bugeaud, Clemens [Akademischer Betreuer] Fuchs e Jörg [Akademischer Betreuer] [Brüdern. "Diophantine Equations and Cyclotomic Fields / Boris Bartolomé. Betreuer: Preda Mihailescu ; Yuri Bilu. Gutachter: Yann Bugeaud ; Clemens Fuchs ; Jörg Brüdern". Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2016. http://d-nb.info/1089093322/34.
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Al-Ghassani, Asma Said Ahmed. "Measures of growth of discrete rational equations". Thesis, Loughborough University, 2010. https://dspace.lboro.ac.uk/2134/6055.
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The general scope of this thesis is aimed at investigating certain classes of discrete equations through the analysis of certain characteristics of the solutions of these equations. We construct new methods of analysis based on the growth of these characteristics that let us single out known integrable discrete equations from certain class of equations. These integrable discrete equations are discrete analogues of the famous Painleve equations.
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Maximenko, Marianna. "Contribution au calcul de la solution générale d'équations en mots". Rouen, 1995. http://www.theses.fr/1995ROUE5003.
Testo completoAbstract (sommario):
Le problème central de la thèse est d'effectuer les recherches du calcul de la solution générale d'équations en mots, donnée sous la forme d'un ensemble fini de collections paramétrées de solutions. Ce projet a été proposé par G. S. Makanin. Nous avons proposé dans ce travail un algorithme quasi linéaire du calcul de la solution générale de l'équation en mots à une variable avec des coefficients. Nous avons décrit la solution générale de l'équation miroir sous la forme des mots à vecteurs. En introduisant la notion d'invariant de Makanin, nous avons obtenu la caractéristique du graphe de l'équation élémentaire. Nous avons aussi introduit les invariants analogues à celui de Makanin pour les transformations élémentaires appliquées aux dernières lettres des membres de l'équation. Les nouvelles notions de variable polarisée, d'atome, de botte d'équations sont introduites. Un nouvel outil pour calculer la solution générale d'équations en mots, basé sur les équations paramétriques formelles, est proposé. Cette méthode est appliquée à l'équation à trois variables. Nous avons obtenu une base pour l'application de cette méthode au cas de quatre variables et nous avons proposé le principe de la classification d'équations paramétriques formelles à l'aide des bottes d'équations. Un algorithme du calcul de la solution générale d'équations et d'inéquations diophantiennes linéaires est présenté de manière non ambiguë
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Neto, Altino da Silva. "Convite às equações diofantinas: uma abordagem para a educação básica". Universidade Federal de Roraima, 2016. http://www.bdtd.ufrr.br/tde_busca/arquivo.php?codArquivo=343.
Testo completoAbstract (sommario):
Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorNesta dissertação, apresentamos os resultados de uma ampla pesquisa bibliográfica sobre as equações diofantinas e seus métodos de solução mais utilizados. A mais simples desta classe de equações é a da forma ax + by = c, com a, b e c números inteiros e ab 6= 0, chamada equação diofantina linear nas duas incógnitas x e y. No trabalho, expomos diversos métodos de resolução destas equações, em duas e três incógnitas. Para tanto, utilizamos conceitos de divisibilidade, divisão euclidiana, máximo divisor comum, números primos, dentre outros, que formam parte do currículo do Ensino Fundamental. No Brasil, as equações diofantinas não são comumente exploradas na Educação Básica, embora sejam perfeitamente compreensíveis nesse nível, como se mostra no texto do professor A. Guelfond, consultado na redação do trabalho. Na dissertação, incluímos, também, um capítulo sobre as contribuições de Diofanto para a Aritmética, que pode ser uma fonte de motivação para o estudo das equações diofantinas; e outro capítulo, ampliando as perspectivas sobre equações diofantinas não lineares. Esperamos que o trabalho seja uma fonte bibliográfica facilmente acessível aos professores da Educação Básica, e estimule seu interesse e criatividade para a introdução elementar desses conteúdos na prática docente e na preparação dos alunos para as Olimpíadas de Matemática.
In this dissertation, the results of a wide bibliographic research about Diophantine equations and their most used solution methods are exposed. The simplest equation of these class is the one in the form ax + by = c, with a, b and c integers numbers and ab 6= 0, called Diophantine linear equation in the unknowns x and y. Divers solutions methods for these equations, in two or three unknowns are discussed. Therefore, concepts like divisibility, Euclidean division, grated common divisor, prime numbers, among others, that are included in the Elementary Schools curriculum. In Brazil, Diophantine equations are not commonly exploited in Basic Education, even though they are perfectly understandable at this educational level, like Professor A. Guelfond shows in his book consulted in the redaction of the dissertation. There are also a chapter about Diophantuss contributions to Arithmetic, which can be a source of motivation to study the Diophantine equations; and another chapter, extending perspectives, about nonlinear Diophantine equations. We hope that the dissertation becomes a suitable easy accessible bibliographic font for Basic Education teachers and stimulates their interest and creativity for an elemental introducing of these contents in their teaching and in the students training for Math Olympiads.
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Souza, Romario Sidrone [UNESP]. "Equações diofantinas lineares, quadráticas e aplicações". Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/149949.
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Este trabalho é resultado de uma pesquisa bibliográfica sobre Diofanto e as equações que levam seu nome, as equações diofantinas. Mais especificamente, apresentamos as equações diofantinas lineares e alguns casos particulares das equações diofantinas quadráticas. Ainda, abordamos um estudo sobre alguns tópicos de teoria dos números e frações contínuas, afim de facilitar o entendimento sobre os teoremas e resultados acerca do tema central deste trabalho.
This work is the result of a bibliographical research about Diophantus and the equations that take his name, the Diophantine equations. More specifically, we present the linear diophantine equations and some particular cases of the quadratic diophantine equations. We have also studied topics about number theory and continuous fractions, in order to facilitate the understanding of theorems and results that are related to the central theme of this work.
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Alves, Lucinda Freese. "Aplicações de equações Diofantinas e um passeio pelo último teorema de Fermat". Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/8104.
Testo completoAbstract (sommario):
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The presente work aims to help students, teachers and lovers of mathematics, to better understand, interpret and solve problems that can be solved through Diophantine Equations. In this way, we present some basic concepts about Diophantine Equations as well as some practical applications. We also discuss Fermat ́s Last Theorem for the cases of n=2, n=3 and n=4, aiming to arouse interest, on the students, in Number Theory.
O presente trabalho tem como objetivo auxiliar estudantes, professores e apaixonados pela matemática, a melhor compreender, interpretar e resolver problemas que possam ser solucionados através das Equações Diofantinas. Desta forma, apresentamos alguns conceitos básicos sobre Equações Diofantinas bem como algumas aplicações práticas. Discutimos ainda, o Último Teorema de Fermat para os casos de n=2, n=3 e n=4, visando despertar o interesse no aluno pela teoria dos números.
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Hanley, Jodi Ann. "Egyptian fractions". CSUSB ScholarWorks, 2002. https://scholarworks.lib.csusb.edu/etd-project/2323.
Testo completoAbstract (sommario):
Egyptian fractions are what we know as unit fractions that are of the form 1/n - with the exception, by the Egyptians, of 2/3. Egyptian fractions have actually played an important part in mathematics history with its primary roots in number theory. This paper will trace the history of Egyptian fractions by starting at the time of the Egyptians, working our way to Fibonacci, a geologist named Farey, continued fractions, Diophantine equations, and unsolved problems in number theory.
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Синиця, О. О. "Нелінійні діофантові рівняння". Thesis, Сумський державний університет, 2018. http://essuir.sumdu.edu.ua/handle/123456789/66829.
Testo completoAbstract (sommario):
Нелінійним діофантовим рівнянням називається рівняння степінь
якого не менший другого. Відмітимо, що загального алгоритму
розв'язування діофантових рівнянь довільного степеня не існує, що
було доведено Матіясевичем Ю. в 1970 році (10-та проблема
Гільберта).
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Burns, Jonathan. "Recursive Methods in Number Theory, Combinatorial Graph Theory, and Probability". Scholar Commons, 2014. https://scholarcommons.usf.edu/etd/5193.
Testo completoAbstract (sommario):
Recursion is a fundamental tool of mathematics used to define, construct, and analyze mathematical objects. This work employs induction, sieving, inversion, and other recursive methods to solve a variety of problems in the areas of algebraic number theory, topological and combinatorial graph theory, and analytic probability and statistics. A common theme of recursively defined functions, weighted sums, and cross-referencing sequences arises in all three contexts, and supplemented by sieving methods, generating functions, asymptotics, and heuristic algorithms.
In the area of number theory, this work generalizes the sieve of Eratosthenes to a sequence of polynomial values called polynomial-value sieving. In the case of quadratics, the method of polynomial-value sieving may be characterized briefly as a product presentation of two binary quadratic forms. Polynomials for which the polynomial-value sieving yields all possible integer factorizations of the polynomial values are called recursively-factorable. The Euler and Legendre prime producing polynomials of the form n2+n+p and 2n2+p, respectively, and Landau's n2+1 are shown to be recursively-factorable. Integer factorizations realized by the polynomial-value sieving method, applied to quadratic functions, are in direct correspondence with the lattice point solutions (X,Y) of the conic sections aX2+bXY +cY2+X-nY=0. The factorization structure of the underlying quadratic polynomial is shown to have geometric properties in the space of the associated lattice point solutions of these conic sections.
In the area of combinatorial graph theory, this work considers two topological structures that are used to model the process of homologous genetic recombination: assembly graphs and chord diagrams. The result of a homologous recombination can be recorded as a sequence of signed permutations called a micronuclear arrangement. In the assembly graph model, each micronuclear arrangement corresponds to a directed Hamiltonian polygonal path within a directed assembly graph. Starting from a given assembly graph, we construct all the associated micronuclear arrangements. Another way of modeling genetic rearrangement is to represent precursor and product genes as a sequence of blocks which form arcs of a circle. Associating matching blocks in the precursor and product gene with chords produces a chord diagram. The braid index of a chord diagram can be used to measure the scope of interaction between the crossings of the chords. We augment the brute force algorithm for computing the braid index to utilize a divide and conquer strategy. Both assembly graphs and chord diagrams are closely associated with double occurrence words, so we classify and enumerate the double occurrence words based on several notions of irreducibility. In the area of analytic probability, moments abstractly describe the shape of a probability distribution. Over the years, numerous varieties of moments such as central moments, factorial moments, and cumulants have been developed to assist in statistical analysis. We use inversion formulas to compute high order moments of various types for common probability distributions, and show how the successive ratios of moments can be used for distribution and parameter fitting. We consider examples for both simulated binomial data and the probability distribution affiliated with the braid index counting sequence. Finally we consider a sequence of multiparameter binomial sums which shares similar properties with the moment sequences generated by the binomial and beta-binomial distributions. This sequence of sums behaves asymptotically like the high order moments of the beta distribution, and has completely monotonic properties.
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Barroso, de Freitas Nuno Ricardo. "Some Generalized Fermat-type Equations via Q-Curves and Modularity". Doctoral thesis, Universitat de Barcelona, 2012. http://hdl.handle.net/10803/91288.
Testo completoAbstract (sommario):
The main purpose of this thesis is to apply the modular approach to Diophantine equations to study some Fermat-type equations of signature (r; r; p) with r >/= 5 a fixed prime and “p” varying. In particular, we will study equations of the form x(r) + y(r) = Cz(p), where C is an integer divisible only by primes “q” is non-identical to 1; 0 (mod “r”) and obtain explicit arithmetic results for “r” = 5, 7, 13.
We start with equations of the form x(5) + y(5) = Cz(p). Firstly, we attach two Frey curves E; F defined over Q(square root 5) to putative solutions of the equation.
Then by using the work of J. Quer on embedding problems and on abelian varieties attached to Q-curves we prove that the p-adic Galois representations attached to E, F can be extended to p-adic representations E), (F) of Gal(Q=Q). Finally, we apply Serre's conjecture to the residual representations (E), (F) and using Siksek's multi-Frey technique we conclude that the initial solution can not exist.
We also describe a general method for attacking infinitely many equations of the form x(r) + y(r) = Cz(p) for all r>/= 7. The method makes use of elliptic curves over totally real fields, modularity and irreducibility results for representations attached to elliptic curves and level lowering theorems for Hilbert modular forms. Indeed, for each fixed “r” we produce several Frey curves defined over K+, the maximal totally real subfield of Q(xi-r). Moreover, if “r” is of the form 6k + 1 we prove the existence of a Frey curve defined over K(0) the subfield of K(+) of degree k. We prove also an irreducibility result for the mod “p” representations attached to certain elliptic curves and a modularity statement for elliptic curves over totally real abelian number fields satisfying some local conditions at 3. Finally, for r = 7 and r = 13 we are able to compute the required spaces of (Hilbert) newforms and by applying our general methods we obtain explicit arithmetic results for equations of signature (7; 7; p) and (13; 13; p).
We end by providing two more Frey k-curves (a generalization of Q-curve), where “k” is a certain subfield of K(+), when “r” is a fixed prime of the form 4m+1.En esta tesis, utilizaremos el método modular para profundizar en el estudio de las ecuaciones de tipo (r; r; p) para r un primo fijado. Empezamos por utilizar la teoría de J. Quer sobre variedades abelianas asociadas con Q-curvas y embedding problems para producir dos curvas de Frey asociadas con hipotéticas soluciones de infinitas ecuaciones de tipo (5; 5; p). Después, utilizando la conjetura de Serre y el método multi-Frey de Siksek demostraremos que las hipotéticas soluciones no pueden existir. Describiremos también un método general que nos permite atacar un número infinito de ecuaciones de tipo (r; r; p) para cada primo “r” mayor o igual que 7. El método hace uso de curvas elípticas sobre cuerpos de números, teoremas de modularidad, teoremas de bajada de nivel y formas modulares de Hilbert. Además, para ecuaciones de tipo (7; 7; p) y (13; 13; p) calcularemos los espacios de formas modulares relevantes y demostraremos que una familia infinita de ecuaciones no admite cierto tipo de soluciones. Además, demostraremos un nuevo teorema de modularidad para curvas elípticas sobre cuerpos totalmente reales abelianos. Finalmente, para primos congruentes con 1 módulo 4 propondremos dos curvas de Frey más. Demostraremos que son “k-curves” (una generalización de Q-curva) y también que satisfacen las propiedades necesarias para que pueda ser útiles en la aplicación del método modular.
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Capilheira, Bianca Herreira. "Equações diofantinas lineares : uma proposta para o Ensino Médio". reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2012. http://hdl.handle.net/10183/62118.
Testo completoAbstract (sommario):
Este trabalho, cuja metodologia foi inspirada na Engenharia Didática, discute e investiga a viabilidade de inserir o ensino/estudo das equações diofantinas lineares no ensino médio. Foi desenvolvida e aplicada uma sequência didática em uma turma do 1º semestre do ensino médio integrado de química do Instituto Federal Sul-Rio-Grandense, Campus Pelotas. Através das atividades executadas pelos alunos, das anotações feitas pela mestranda e da filmagem de todas as aulas, foi possível coletar os dados sobre toda a experiência. Esta foi iniciada e baseada em um jogo nomeado “escova diofantina”, derivado do jogo “escova”, seguido de atividades estruturadas com exercícios, questionamentos e debates que encaminharam os alunos, de forma natural, para a construção e estudo do conteúdo desejado. Elaboramos a sequência didática com objetivos bem definidos em cada atividade. Após o término das aulas, analisamo-las e reformulamo-las. Assim, no Apêndice B, apresentamos uma proposta de sequência didática renovada e pronta para ser aplicada por qualquer professor interessado em lecionar equações diofantinas no ensino médio. Os resultados das análises dos dados indicaram que os alunos do primeiro ano do ensino médio apresentam plenas condições matemáticas para a compreensão e construção dos conceitos e propriedades básicas relacionadas às equações diofantinas lineares.This work, whose methodology is inspired by didactical engineering, discusses and investigates the viability of introducing linear diophantine equations at High School level study and teaching. We developed and applied a didactical sequence to a first semester chemistry oriented high school at the Pelotas campus of the Sul-Rio-Grandense Federal Institute. We collected the data of this whole experience, starting with all the activities performed by the students and continuing with notes taken by the author as well as the whole class footage. We started the seminars with a card game that we called “diophantine escova”, derived from the usual “escova” card game. We followed it by structured activities with exercises and several debates that led the students, in a natural way, to understand the definitions, concepts and results about Diophantine Equations. The didactical sequence we have created had very clear and specific goals in each activity. When the seminars ended, we analyzed and reformulated the sequence and therefore, in Appendix C, we present a totally improved and ready to use sequence for any teacher interested in developing linear diophantine equations in high school. The data analysis indicated that fist year high school students have the necessary mathematical skills to understand all concepts and results of basic linear diofantine equations.
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Silva, Filardes de Jesus Freitas da. "Equações diofantinas classicas e aplicações". [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307049.
Testo completoAbstract (sommario):
Orientador: Emerson Alexandre de Oliveira LimaDissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho focalizamos os principais conceitos da teoria elementar dos números objetivando uma melhor compreensão das Equações Diofantinas Clássicas e suas aplicações e para isto explicitamos os conceitos de Números primos, Algoritmo de Euclides, Máximo divisor comum e Mínimo múltiplo comum, assim como a teoria das Congruências, uma abordagem sobre a Criptografica RSA e Soma de Inteiros. Palavras-Chave: Congruências Lineares, Soma de Inteiros, Equação de Fermat, Soma de Quadrados
Abstract: In this work we focus the main concepts of the elementary theory of numbers seeking a better understanding of Classical diophantine equations and their applications for this and explained the concepts of prime numbers, algorithms of Euclid, maximum common divisor and least common multiple and the theory of congruence , an approach on the RSA encryption and Sum of Integers. Keywords: Linear congruence, Sum of Integers, equation of Fermat, Sum of Squares
Mestrado
Teoria dos Numeros
Mestre em Matemática
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Пальчик, О. О. "Комп'ютерна реалізація методів рішення діофантових рівнянь". Master's thesis, Сумський державний університет, 2018. http://essuir.sumdu.edu.ua/handle/123456789/72319.
Testo completoAbstract (sommario):
Реалізовано програмний засіб для рішення діофантового рівняння, розроблений на мові C# в середовищі програмування Microsoft Visual Studio Community 2017. Функціональне моделювання програмного засобу виконано на базі технології SADT. Проектування програмного продукту виконано в нотації UML.
35
Khoshnoudirad, Daniel. "Aspects combinatoires des motifs linéaires en géométrie discrète". Thesis, Paris Est, 2016. http://www.theses.fr/2016PESC1046.
Testo completoAbstract (sommario):
La Géométrie Discrète, comme Science de l'Informatique Théorique, étudie notamment les motifs linéaires tels que les primitives discrètes apparaissant dans les images : les droites discrètes, les segments discrets, les plans discrets, les morceaux de plans discrets par exemple. Dans ce travail, je me concentre tout particulièrement sur les diagrammes de Farey qui apparaissent lors de l'étude des primitives discrètes que sont les (m,n)-cubes, autrement dit les morceaux de plans discrets. J’étudie notamment la Combinatoire des droites formant les diagrammes de Farey, en établissant des formules exactes. Je montre alors que certaines méthodes utilisées auparavant ne permettront pas d'optimiser la Combinatoire des (m,n)-cubes. J'obtiens aussi une estimation asymptotique en utilisant la Théorie des Nombres Combinatoire. Puis, concernant les sommets apparaissant dans les diagrammes de Farey, j'obtiens une borne inférieure. J'analyse alors les stratégies déjà mises en place pour l'étude des $(m,n)$-cubes par les seuls diagrammes de Farey en deux dimensions. Afin d'obtenir de nouvelles bornes plus précises pour les $(m,n)$-cubes, une des seules méthodes actuellement existantes, est de proposer une généralisation de la notion de pré image d'un segment discret, à celle de pré image d'un $(m,n)$-cube, avec pour conséquence une nouvelle inégalité combinatoire sur le cardinal des (m,n)-cubes (inégalité qui pourrait même s'avérer être une égalité). Ainsi, nous introduisons la notion de diagramme de Farey en trois dimensionsDiscrete Geometry, as Theoretical Computer Science, studies in particular linear patterns such as discrete primitives in images: the discrete lines, discrete segments, the discrete planes, pieces of discrete planes, for example. In this work, I particularly focused on Farey diagrams that appear in the study of the $ (m, n) $ - cubes, ie the pieces of discrete planes. Among others, I study the Combinatorics of the Farey lines forming diagram Farey, establishing exact formulas. I also get an asymptotic estimate using Combinatorial Number Theory. Then, I get a lower bound for the cardinality of the Farey vertices. After that, we analyze the strategies used in the literature for the study of (m, n)- cubes only by Farey diagrams in two dimensions. In order to get new and more accurate bounds for (m, n)- cubes, one of the few available methods, is to propose a generalization for the concept of preimage of a discrete segment for (m, n) - cube, resulting in a new combinatorial inequality. Thus, we introduce the notion Farey diagram in three dimensions
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Campos, Adilson de. "EQUAÇÕES DIOFANTINAS LINEARES: POSSIBILIDADES DIDÁTICAS USANDO A RESOLUÇÃO DE PROBLEMAS". Universidade Federal de Santa Maria, 2015. http://repositorio.ufsm.br/handle/1/10945.
Testo completoAbstract (sommario):
Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorThis work presents an educational experiment carried out in a 9th grade class of elementary school, in order to assess the didactic and pedagogical possibilities involving the Linear Diophantine Equations theme, with the contextual support of Problem Solving. This application intends to expand the students' conceptions in arithmetic and algebra courses, also providing a concrete possibility of applicability of the greatest common divisor of two integers, a very neglected theme throughout the elementary school. In a level of elementary school, one of the main vehicles that allows you to work the initiative, creativity and exploring spirit is through Problem Solving. A Mathematics Teacher has a great opportunity to challenge the curiosity of the students by presenting them problems that are compatible with their knowledge and guiding them through incentive questions and this teacher can also try to input on them a taste for discovery and independent thinking. Thus, a very reasonable way is to prepare the student to deal with new situations, whatever they may be. The paper is organized in three chapters. In the first chapter entitled "Problem Solving in mathematics teaching" a theoretical foundation on the Teaching of Problem Solving is searched based on the Hungarian-American author George Polya and Luiz Roberto Dante and, it also presents some aspects from the learning theory proposed by Vygotsky. In the second chapter entitled "arithmetic concepts" the themes treated are: Greatest Common Divisor (gcd), Euclidean algorithm, Bèzout theorem and Linear Diophantine Equations. In the third and final chapter entitled "pedagogical experimentation" as mentioned above, the experimentation in a class of ninth grade of an elementary school. This experiment is based on the Didactic Engineering methodology, comprising the following stages: theme and scope of action; previous analyzes associated with the dimensions: epistemological, didactic and cognitive; prior analysis; experimentation; aftermost analysis and validation of Didactic Engineering.
Este trabalho apresenta uma experimentação pedagógica realizada numa turma de 9ºano do Ensino Fundamental com o objetivo de aferir as possibilidades didático-pedagógicas envolvendo a temática Equações Diofantinas Lineares, tendo como suporte contextual a Resolução de Problemas. Tal aplicação tem o intento de ampliar as concepções dos alunos nos campos da aritmética e da álgebra, dando também uma possibilidade concreta de aplicabilidade do máximo divisor comum de dois números inteiros, tema tão negligenciado ao longo do Ensino Fundamental. Em um nível de Ensino Fundamental, um dos principais veículos que permite trabalhar a iniciativa, a criatividade e o espírito explorador é a Resolução de Problemas. O professor de Matemática tem, dessa forma, uma grande oportunidade de desafiar a curiosidade de seus alunos, apresentando-lhes problemas compatíveis com os conhecimentos destes e orientando-os através de indagações incentivadoras, podendo incutir-lhes o gosto pela descoberta e pelo raciocínio independente. Assim, um caminho bastante razoável é preparar o aluno para lidar com situações novas, quaisquer que sejam elas. O trabalho está organizado em três capítulos. No primeiro capítulo intitulado A Resolução de Problemas no ensino da Matemática busca-se uma fundamentação teórica sobre a Didática da Resolução de Problemas no autor húngaro-americano George Polya e Luiz Roberto Dante e, também, são apresentados alguns aspectos da teoria da aprendizagem proposta por Vygotsky. No segundo capítulo intitulado conceitos de aritmética são tratados os temas: Máximo Divisor Comum (mdc), Algoritmo de Euclides, Teorema de Bèzout e Equações Diofantinas Lineares. No terceiro e último capítulo intitulado experimentação pedagógica é apresentada a experimentação supracitada numa turma de nono ano do Ensino Fundamental. Tal experimentação é baseada na metodologia Engenharia Didática, compreendendo os seguintes momentos: tema e campo de ação; análises prévias associadas às dimensões: epistemológica, didática e cognitiva; análise a priori; experimentação; análise a posteriori e validação da Engenharia Didática.
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Nascimento, NatÃlia Medeiros do. "EquaÃÃes diofantinas e o mÃtodo das secantes e tangentes de Fermat". Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=12409.
Testo completoAbstract (sommario):
CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel SuperiorAo longo das Ãltimas dÃcadas, a transmissÃo do conhecimento matemÃtico na EducaÃÃo BÃsica sofreu diversas mudanÃas. âO Ensino Tradicionalâ da matemÃtica era baseado na memorizaÃÃo de fÃrmulas, havendo assim uma mecanizaÃÃo no processo de resoluÃÃo de problemas, onde o discente era visto como um ser passivo. A nova visÃo de ensino, que busca significar o que conteÃdo exposto em sala, motivou a escolha desse tema, visto que situaÃÃes problemas envolvendo equaÃÃes diofantinas podem ser facilmente percebidas em nosso cotidiano. O objetivo deste trabalho à oportunizar a realizaÃÃo de uma leitura consultiva para o professor do Ensino BÃsico, e asseverar que essas equaÃÃes podem ser aplicadas na EducaÃÃo BÃsica como uma ferramenta que instiga o pensamento lÃgico, o raciocÃnio, a compreensÃo e a interpretaÃÃo matemÃtica. A formulaÃÃo desse material que està dividido em cinco capÃtulos se deu atravÃs de levantamento bibliogrÃfico por meio de pesquisas descritivas. A introduÃÃo compÃe o primeiro capÃtulo. O segundo capÃtulo versa sobre o Legado de Diofanto: vida e obras, ressaltando sua obra titulada âArithmeticaâ que contribuiu significativamente para o desenvolvimento da teoria dos nÃmeros. O terceiro capÃtulo trata das equaÃÃes diofantinas lineares de n variÃveis. O quarto capÃtulo aborda as ternas itagÃricas, o MÃtodo das Secantes e Tangentes de Fermat na busca de soluÃÃes racionais para quaÃÃes, com coeficientes racionais, da forma ax2+by2 = c, e um caso particular do Ãltimo Teorema de Fermat. O quinto capÃtulo à composto de problemas sobre equaÃÃes diofantinas lineares.
Over the past decades, the transmission of mathematical knowledge in basic education has undergone several changes. The âTeaching Traditionalâ math was based on memorizing formulas, so there mechanization in problem solving where the student was seen as a liability to be process. The new vision of education that seeks to signify exposed to room content, motivated the choice of this theme, as diophantine equations involving situations problems can be easily noticed in our daily lives. The objective of this work is an opportunity for a realization of an advisory reading for the teacher of basic education, and assert that these equations can be applied in basic education as a tool that encourages the logical thinking, reasoning, understanding and mathematical interpretation. The formulation of this material which is divided into five chapters was through literature review through descriptive research. The introduction comprises the first chapter. The second chapter deals with the Legacy of Diophantus: life and works, emphasizing his work entitled âArithmeticaâ which contributed significantly to the development of number theory. The third chapter deals with linear Diophantine equations in n variables. The fourth chapter discusses the Pythagorean tender, Fermatâs of secants and Tangents method, in finding rational solutions to equations with rational coefficients, of the form ax2 + by2 = c and a particular case Fermatâs Last Theorem. The fifth chapter is composed of problems on linear diophantine equations.
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Dantas, Joice de Andrade. "De solutione problematum diophanteorum per n?meros integros : o primeiro trabalho de Euler sobre equa??es diofantinas". Universidade Federal do Rio Grande do Norte, 2011. http://repositorio.ufrn.br:8080/jspui/handle/123456789/14500.
Testo completoAbstract (sommario):
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Previous issue date: 2011-11-07The present dissertation analyses Leonhard Euler?s early mathematical work as Diophantine Equations, De solutione problematum diophanteorum per n?meros ?ntegros (On the solution of Diophantine problems in integers). It was published in 1738, although it had been presented to the St Petersburg Academy of Science five years earlier. Euler solves the problem of making the general second degree expression a perfect square, i.e., he seeks the whole number solutions to the equation ax2+bx+c = y2. For this purpose, he shows how to generate new solutions from those already obtained. Accordingly, he makes a succession of substitutions equating terms and eliminating variables until the problem reduces to finding the solution of the Pell Equation. Euler erroneously assigns this type of equation to Pell. He also makes a number of restrictions to the equation ax2+bx+c = y and works on several subthemes, from incomplete equations to polygonal numbers
Nesta pesquisa analisamos historicamente e matematicamente o primeiro trabalho de Leonhard Euler sobre Equa??es Diofantinas o De solutione problematum diophanteorum per n?meros integros ( Sobre a solu??o de problemas diofantinos por n?meros inteiros ). Foi publicado em 1738, embora apresentado ? Academia de S?o Petersburgo cinco anos antes. No texto, Euler trata do problema de fazer com que a express?o generalizada do segundo grau seja igual a um quadrado perfeito, isto ?, procura solu??es no conjunto dos n?meros inteiros para equa??o ax2+bx+c = y2. Para tanto, Euler mostra como descobrir mais solu??es depois que uma primeira ? encontrada, fazendo uma s?rie de substitui??es combinando termos e eliminando vari?veis, at? que o trabalho se resume a encontrar a solu??o para ,q=ⱱap?+1 uma equa??o de Pell. Este trabalho ? o primeiro tamb?m em que Euler atribui erroneamente esse tipo de equa??o a Pell. Euler faz tamb?m, uma s?rie de restri??es para a equa??o ax2+bx+c = y2 e trabalha com diversos subcasos, que v?o desde equa??es incompletas at? o trabalho com n?meros poligonais
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Alcántara, Bode Julio. "Some Properties of the Beurling Correlation Function". Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/97055.
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PRAZERES, Sidmar Bezerra dos. "O Teorema chinês dos restos e a partilha de senhas". Universidade Federal Rural de Pernambuco, 2014. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/6709.
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This paper aims to show the reader the importance of some topics of Number Theory. Work here, and prerequisites (Euclid Algorithms, Divisibility, Maxim Common Divisor), content with Linear Diophantine equations, congruences, and the main theme, which is the mighty Chinese Remainder Theorem of presenting their theories, importance, applicability on the day and its usefulness in the Theory of Numbers. The main applicability of Chinese Remainder Theorem of this work is Sharing Passwords. Sharing of passwords is a security mechanism, where a certain amount of people take possession of a key to access the secret without the possibility of obtaining the secret with his own key.
Este trabalho tem como objetivo mostrar ao leitor a importância de alguns t ópicos da Teoria dos N úmeros. Trabalharemos aqui, al ém de pré-requisitos (Algoritmo de Euclides, Divisibilidade, M áximo Divisor Comum), conte údos como Equa ções Diofantinas Lineares, Congruências e o principal tema, que e o poderoso Teorema Chinês dos Restos, apresentando suas teorias, importâncias, aplicabilidade no dia a dia e sua a utilidade na Teoria dos N úmeros. A principal aplicabilidade do Teorema Chinês apresentada neste trabalho e a Partilha de Senhas. Esta partilha de senhas é um mecanismo de seguran ça, onde uma certa quantidade de pessoas tomam posse de uma chave de acesso sem a possibilidade de obter a senha principal com a sua pr ópria chave.
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Weeman, Glenn Steven. "A Diophantine Equation for the Order of Certain Finite Perfect Groups". University of Akron / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=akron1396902470.
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Haristoy, Julien. "Equations diophantiennes exponentielles". Université Louis Pasteur (Strasbourg) (1971-2008), 2003. http://www.theses.fr/2003STR13123.
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Nguyen, Phu Qui Pierre. "Equations de Mahler et hypertranscendance". Paris 6, 2012. http://www.theses.fr/2012PA066809.
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Soit K un corps équipé d'un endomorphisme \sigma. Dans cette thèse, nous montrons que la théorie de Galois aux \sigma-différences bien connue dans le cas où \sigma est un automorphisme du corps K peut être adaptée au cas où \sigma n'est plus nécessairement surjectif, en passant à la clôture inversive de K. Nous utilisons ensuite cette théorie de Galois pour donner un critère d'indépendance algébrique pour les solutions de \sigma-équations du premier ordre. Ce résultat nous permet de caractériser les solutions hyperalgébriques de ces \sigma-équations lorsque K est muni d'une dérivation vérifiant une hypothèse de quasi-commutation avec \sigma. En appliquant notre critère d'indépendance algébrique à l'opérateur de Mahler, nous donnons enfin une preuve galoisienne d'un théorème d'hypertranscendance de Ke. NishiokaLet K be a field equipped with an endomorphism \sigma. In this thesis, we show that the Galois theory for \sigma-difference equations, well known if \sigma is an automorphism of K, can be adapted to the case when \sigma is not necessarily surjective anymore, by passing to the inversive closure of K. We then use this Galois theory to give an algebraic independence criterion for solutions of first order \sigma-equations. This result allows us to characterize the hyperalgebraic solutions of such \sigma-equations when K is endowed with a derivation which almost commutes with \sigma. Applying our algebraic independence criterion to the Mahler operator setting, we give a galoisian proof of a hypertranscendence theorem of Ke. Nishioka
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Borges, Fábio Vieira de Andrade. "Equações diofantinas lineares em duas incógnitas e suas aplicações". Universidade Federal de Goiás, 2013. http://repositorio.bc.ufg.br/tede/handle/tede/3124.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The main objective of this assignment is to help students and also teachers with the resolution and understanding of problems involving the Linear Diophantine Equations with Two Incognits through the elaboration and application of didactic activities in order to contribute to the study of this kind of equations. Through the tasks it was aimed to dothe integration of Arithmetic with Algebra and Geometry by using some computational programs which worked as support to the graphical visualization of the entire solutions. In the first chapters the essence of the Elementary Theory of Numbers will be better known, since the mathematical tools which will be used to solve linear Diophantine equations will be displayed and demonstrated, some of them already known, like the greatest common divisor (g.d.c). Then the Diophantine equations and theirapplication methods for the solution of daily problems will be introduced. The Conclusion of this study highlights the importance of algebraic and geometric interpretation of Linear Diophantine Equations, and also emphasizes that the contact with problems of this area contributes to the students reasoning abilities development in a creative way. It is important to emphasize that this issue can be introduced in high school.
O presente trabalho tem como objetivo principal auxiliar os alunos e professores na resolução e compreensão de problemas envolvendo as Equações Diofantinas Lineares com Duas Incógnitas através da elaboração e aplicação de atividades didáticas destinadas a contribuir para o estudo desse tipo de equações. Procurou-se nas tarefas fazer a integração da Aritmética com a Álgebra e a Geometria, utilizando-se de alguns programas computacionais que serviram de suporte para as visualizações gráficas das soluções inteiras. Nos primeiros capítulos vamos conhecer melhor a essência da Teoria Elementar dos Números, pois apresentaremos e demonstraremos as ferramentas matemáticas que serão utilizadas na resolução das Equações Diofantinas Lineares, algumas delas já conhecidas, que é o caso do máximo divisor comum (m.d.c). Em seguida serão introduzidas as equações diofantinas e os métodos de determinação de soluções da mesma para aplicação em resolução de problemas do cotidiano. A conclusão desse trabalho ressalta a importância da interpretação algébrica e geométrica das Equações Diofantinas Lineares, e que o contato com problemas desta área contribui para que o aluno desenvolva, de forma criativa suas habilidades de raciocínio. É importante enfatizar que esse tema pode ser abordado no Ensino Médio.
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Hanine, El Mostafa. "Equations diophantiennes p-adiques et congruences modulo p2". Toulouse 3, 1990. http://www.theses.fr/1990TOU30088.
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On demontre que pour tout entier naturel d, il existe un plus petit entier p(d) tel que si p est un nombre premier superieur ou egal a p(d) et f,z#px#1. . . , x#2#d#+#1 sans terme constant de degre d l'equation f(x#1. . . , x#2#d#+#1)0(p#2) admet une solution primitive. On demontre aussi que p(2)=2p(3)=3,p(d)>2 si d4 et p(d)>p si d est multiple de p#2p avec p=2
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Rahimi, Shahriar. "A NOVEL LINEAR DIOPHANTINE EQUATION-BAESD LOW DIAMETER STRUCTURED PEER-TO-PEER NETWORK". OpenSIUC, 2017. https://opensiuc.lib.siu.edu/dissertations/1462.
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This research focuses on introducing a novel concept to design a scalable, hierarchical interest-based overlay Peer-to-Peer (P2P) system. We have used Linear Diophantine Equation (LDE) as the mathematical base to realize the architecture. Note that all existing structured approaches use Distributed Hash Tables (DHT) and Secure Hash Algorithm (SHA) to realize their architectures. Use of LDE in designing P2P architecture is a completely new idea; it does not exist in the literature to the best of our knowledge. We have shown how the proposed LDE-based architecture outperforms some of the most well established existing architecture. We have proposed multiple effective data query algorithms considering different circumstances, and their time complexities are bounded by (2+ r/2) only; r is the number of distinct resources. Our alternative lookup scheme needs only constant number of overlay hops and constant number of message exchanges that can outperform DHT-based P2P systems. Moreover, in our architecture, peers are able to possess multiple distinct resources. A convincing solution to handle the problem of churn has been offered. We have shown that our presented approach performs lookup queries efficiently and consistently even in presence of churn. In addition, we have shown that our design is resilient to fault tolerance in the event of peers crashing and leaving. Furthermore, we have proposed two algorithms to response to one of the principal requests of P2P applications’ users, which is to preserve the anonymity and security of the resource requester and the responder while providing the same light-weighted data lookup.
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Синиця, О. "Методи розв'язування діофантових рівнянь". Thesis, Cумський державний університет, 2016. http://essuir.sumdu.edu.ua/handle/123456789/48885.
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Розв’язок рівнянь в цілих числах є однією з стародавніх
математичних задач. Основне джерело, яке дійшло до наших часів –
видання праці Діофанта «Арифметика». На жаль, з тринадцяти книг,
що входили до цього видання, тільки шість збереглися до Середніх
віків, саме вони і стали джерелом натхнення для багатьох
математиків.
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Abouzaid, Mourad. "Aspects effectifs d'analyse diophantienne". Bordeaux 1, 2006. http://www.theses.fr/2006BOR13196.
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Cette thèse traite de trois problèmes diophantiens distincts. Le premier chapitre est consacré à l’étude des diviseurs premiers de suites d’entiers connues sous le nom de nombres de Lucas et des nombres de Lehmer. Un théorème de Yu. BILU, G. HANROT et P. M. VOUTIER nous assure qu’à partir d’un certain rang (indépendent de la suite choisie, et explicite), tout nombre de Lucas (resp. De Lehmer ) admet un “nouveau” diviseur premier. Dans ce premier chapitre, nous complétons la liste des cas pathologiques commencée par P. M. VOUTIER d’une part et Yu. BILU, G. HANROT et P. M. VOUTIER d’autre part. Le second chapitre est pr´eparatoire. Dans celui-ci, nous mettons en place deux outils fondamentaux pour la résolution effective de certaines classes d’équations diophantiennes : les hauteurs de Weil et les séries de Puiseux. Dans le troisième chapitre, on s’intéresse aux équations algébriques du type F(x, y) = 0 o`u le polynôme F vérifie F(0, 0) = 0. Un théorème de Skolem montre que ces équations n’admettent qu’un nombre fini de solutions entières à pgcd fixé. Après avoir généralisé la notion de pgcd aux nombres algébriques, nous établissons un théorème effectif donnant la taille maximale des solutions algébriques des équations du type Skolem. Enfin, le quatrième et dernier chapitre de cette thèse est consacré à l’étude des équations de Thue généralisées. Celui-ci a en 1909 que si F(X, Y ) était une forme homogène à coefficients entiers, irréductible sur Q et de degré n # 3, alors les équations F(x, y) = A pour A 2 Z\{0} n’admettaient chacune qu’un nombre fini de solutions entières. La encore, nous nous sommes attaché à généraliser le problème initial en considérant les équations du type NL/K(F(x, y)) = A où K est un corps de nombres donné, L est une extension finie de K de degré n # 3, NL/K est la norme relative de L sur K, F est un polynôme à coefficients dans L et A 2 K#. Nous établissons encore une fois une borne effective pour la taille des solutions S-entières de ce type d’équations. Ce dernier chapitre est le fruit d’un travail effectué conjointement avec A. Berczes et Yu. Bilu.
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Dehghan, Nezhad Akbar. "Equations cohomologiques de flots riemanniens et de difféomorphismes d'Anosov". Phd thesis, Université de Valenciennes et du Hainaut-Cambresis, 2006. http://tel.archives-ouvertes.fr/tel-00145138.
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Résumé: Dans cette thèse, on étudie les équations cohomologiques discrètes et continues dans les situations qui suivent.1- Pour un champ de vecteurs X qui définit un feuilletage riemannien complet sur une variété M, on donne explicitement les conditions qui permettent de résoudre complètement l'équation cohomologique continue.
2- Pour un champ X sur une la variété M obtenus (M et X) par suspension d'un difféomorphisme γ : N → N, on montre que l'équation cohomologique discrète du système dynamique discret (N, γ) est équivalente à l'équation cohomologique continue du système dynamique continu (M, X) .
3- Dans le cas où la variété M est le quotient TAⁿ⁺¹du groupe de Lie G=ℝⁿ ⋊A ℝ par le réseau Γ=ℤⁿ ⋊Aℤ avec A ∈SL(n,ℤ)
hyperbolique à valeurs propres réelles positives et X l'élément de l'algèbre de Lie Ģ de G qui induit le flot d'Anosov ℱ sur TAⁿ⁺¹,on donne explicitement les solutions des deux équations en question ainsi que d'autres invariants géométriques qui leur sont associés notamment la cohomologie feuilletée de ℱ et les distrbutions A-invariantes
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Валенкевич, М. Є. "Діофантові рівняння". Thesis, Сумський державний університет, 2014. http://essuir.sumdu.edu.ua/handle/123456789/38857.
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