Spelling suggestions: "subject:"diophantine equations"" "subject:"diophanitine equations""
1 
De quibusdam aequationibus indeterminatis quarti gradusKramer, August Ephraim. January 1839 (has links)
Thesis (doctoral)FriedrichWilhelmsUniversität Berlin, 1839. / Vita.

2 
An exploration of Diophantine equationsHoffmaster, Christina Diane 27 November 2012 (has links)
This paper outlines recent research on Diophantine equations. The topics discussed include methods for generating solutions to specific equations and analysis of patterns in solutions to some specific equations / text

3 
Linear diophantine equations: integration of disaggregation with LLL algorithm.January 2014 (has links)
線性丢番圖方程系統（LDEs）連同其子類子集和問題在現實世界有大量且重要的應用。可是當線性番圖方程系統的整數解集被限定在一個有界的多面體中，則此系統屬于NP類問題。同理其子類子集和問題也同屬于NP類問題。另一方面，密度（density）接近或等于一的子集和問題已在文獻中被證實爲最難的一類子集和問題，並且現有的針對此最難子集和問題的所有解決方案都不能達到令人滿意的成功率。因此，在這篇論文中，我們旨在提出有效的算法來求解線性番圖方程系統以及其子類問題密度爲一的子集和問題。 / 我們在此論文中的研究包括：1）基于格理論和LLL算法的性質，采用並改良針對LDEs的格表達式（lattice formulations）；2）提出針對子集和問題的分解（disaggregation）技術；3）創造性地將分解技術與格表達式整合在一起，從而提高求解密度爲一的子集和問題的成功率。 / 數值實驗顯示，我們提出的新整合算法對提高密度爲一的子集和問題的成功率有著顯著的效果。比如，針對維數分別爲20,30和40的密度爲一的子集和問題，對各個維數隨機産生的100個問題，我們的新整合算法均可將成功率提高到100%。同時，針對新整合算法的理論分析顯示，能將短且非01的整數解切割掉的分解在達到新整合算法的顯著實驗效果中起到了關鍵作用。 / While systems of linear Diophantine equations (LDEs) with bounded feasible set, including subset sum problem as its special subclass, find wide, and often significant, realworld applications, they unfortunately belong to the NP class in general. Furthermore, the literature has revealed that subset sum problems with their density close to one constitute the hardest subclass of subset sum problems and all the existing solution methods do not perform to a satisfactory level (with low success ratio) even when the problem size is only medium. / We take the challenge in this thesis to investigate lattice formulations for systems of LDEs in which LLL basis reduction algorithm (LLL algorithm) is utilized, propose disaggregation techniques for subset sum problems, and develop a powerful integration of disaggregation techniques with lattice formulations in solving feasible subset sum problems. / More specifically, the contributions in this thesis can be classified into three parts: i) we propose two revised lattice formulations of Aardal et al. (2000) for systems of LDEs to enhance further the computational capability of the LLL algorithm; ii) we study properties related to disaggregation of a single LDE and investigate thoroughly disaggregation schemes based on modular transformations; and iii) we develop a novel version of LLL algorithm by integrating modular disaggregation into the solution process. Promising numerical results have been achieved when applying our newly proposed LLL algorithm in tackling hard subset sum problems with density close to one. For instance, the success ratio can be raised to 100% for 100 randomly generated hard subset sum problems with dimensions 20, 30, and 40, respectively. We carry out theoretical study for possible driving force behind the success of our new algorithm, including dimension reduction of the solution space, information recovering of LDEs, and mechanism in cutting off short nonbinary integer solutions when attaching disaggregation with LLL algorithm. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lu, Bojun. / Thesis (Ph.D.) Chinese University of Hong Kong, 2014. / Includes bibliographical references (leaves 143153). / Abstracts also in Chinese.

4 
Exploring calculus students' understanding of L'Hôpital's RuleBeauchamp, Bradley K. McCrone, Sharon. Rich, Beverly Susan. January 2006 (has links)
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 (cochairs), James F. Cottrill, Lucian L. Ionescu. Includes bibliographical references (leaves 155159) and abstract. Also available in print.

5 
A discussion of homogenous quadratic equationsKaminski, Lance January 1900 (has links)
Master of Science / Department of Mathematics / Christopher G. Pinner / This thesis will look at Quadratic Diophantine Equations. Some well known proofs, including how to compute all Pythagorean triples and which numbers can be represented by the sum of two and four squares will be presented. Some concepts that follow from these theorems will also be presented. These include how to compute all Pythagorean Quadruples, which number can be represented by the difference of two squares and the Crossed Ladders problem. Then, Ramanujan's problem of finding which positive integers, a,b,c and d which allow aw^2+bx^2+cy^2+dz^2 to represent all natural numbers will be shown. The paper will conclude with a lengthy discussion of Uspensky's proof on which numbers can be represented by the sum three squares.

6 
Diophantine equations with arithmetic functions and binary recurrences sequencesFaye, Bernadette January 2017 (has links)
A thesis submitted to the Faculty of Science, University of the
Witwatersrand and to the University Cheikh Anta Diop of Dakar(UCAD)
in fulfillment of the requirements for a Dualdegree for Doctor in
Philosophy in Mathematics. November 6th, 2017. / This thesis is about the study of Diophantine equations involving binary recurrent
sequences with arithmetic functions. Various Diophantine problems are investigated
and new results are found out of this study. Firstly, we study several
questions concerning the intersection between two classes of nondegenerate binary
recurrence sequences and provide, whenever possible, effective bounds on
the largest member of this intersection. Our main study concerns Diophantine
equations of the form '(jaunj) = jbvmj; where ' is the Euler totient function,
fungn 0 and fvmgm 0 are two nondegenerate binary recurrence sequences and
a; b some positive integers. More precisely, we study problems involving members
of the recurrent sequences being repdigits, Lehmer numbers, whose Euler’s
function remain in the same sequence. We prove that there is no Lehmer number
neither in the Lucas sequence fLngn 0 nor in the Pell sequence fPngn 0. The
main tools used in this thesis are lower bounds for linear forms in logarithms
of algebraic numbers, the socalled BakerDavenport reduction method, continued
fractions, elementary estimates from the theory of prime numbers and sieve
methods. / LG2018

7 
Rational and Heron TetrahedraChisholm, Catherine Rachel January 2004 (has links)
Rational tetrahedra are tetrahedra with rational edges. Heron tetrahedra are tetrahedra with integer edges, integer faces areas and integer volume  the threedimensional analogue of Heron triangles. Of course, if a rational tetrahedron has rational face areas and volume then it is easy to scale it up to get a Heron tetrahedron. So we also use `Heron tetrahedra' when we mean tetrahedra with rational edges, areas and volume. The work in this thesis is motivated by Buchholz's paper {\it Perfect Pyramids} [4]. Buchholz examined certain configurations of rational tetrahedra, looking first for tetrahedra with rational volume, and then for Heron tetrahedra. Buchholz left some of the cases he examined unsolved and Chapter 2 is largely devoted to the resolution of these. In Chapters 3 and 4 we expand upon some of Buchholz's results to find infinite families of Heron tetrahedra corresponding to rational points on certain elliptic curves. In Chapters 5 and 6 we blend the ideas of Buchholz in [4] and of Buchholz and MacDougall in [7], and consider rational tetrahedra with edges in arithmetic (AP) or geometric (GP) progression. It turns out that there are no Heron AP or GP tetrahedra, but AP tetrahedra can have rational volume. They can also have one rational face area, although only one AP tetrahedron has been found with a rational face area and rational volume. For GP tetrahedra there are still unsolved cases, but we show that if GP tetrahedra with rational volume exist, then there are only finitely many. The faces of a rational GP tetrahedron are never rational. Much of the work in these two chapters also appeared in the author's Honours thesis, but has been refined and extended here, and is included to give a more complete picture of the work on Heron tetrahedra which has been done to date. In the final chapter we use a different approach and concentrate on the face areas first, instead of the volume. To make it easier (hopefully) to find tetrahedra with all faces having rational area, we place restrictions on the types of faces and number of different faces the tetrahedra have. / Masters Thesis

8 
Multiplicities of Linear Recurrence SequencesAllen, Patrick January 2006 (has links)
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 zeromultiplicity of a linear recurrence sequence of order <em>t</em> by a function involving <em>t</em> alone. Moreover we improve on Schmidt's bound by making some minor changes to his argument.

9 
Multiplicities of Linear Recurrence SequencesAllen, Patrick January 2006 (has links)
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 zeromultiplicity of a linear recurrence sequence of order <em>t</em> by a function involving <em>t</em> alone. Moreover we improve on Schmidt's bound by making some minor changes to his argument.

10 
On effective irrationality measures for some values of certain hypergeometric functionsHeimonen, A. (Ari) 20 March 1997 (has links)
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 nonarchimedean 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.

Page generated in 0.7112 seconds