Dissertations / Theses: 'Prime number'

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Ho, Kwan-hung, and 何君雄. "On the prime twins conjecture and almost-prime k-tuples." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2002. http://hub.hku.hk/bib/B29768421.

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Buchanan, Dan Matthews. "Analytic Number Theory and the Prime Number Theorem." Youngstown State University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1525451327211365.

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Hendi, Yacoub. "On The Prime Number Theorem." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-441288.

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4

Anderson, Crystal Lynn. "An Introduction to Number Theory Prime Numbers and Their Applications." Digital Commons @ East Tennessee State University, 2006. https://dc.etsu.edu/etd/2222.

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The author has found, during her experience teaching students on the fourth grade level, that some concepts of number theory haven't even been introduced to the students. Some of these concepts include prime and composite numbers and their applications. Through personal research, the author has found that prime numbers are vital to the understanding of the grade level curriculum. Prime numbers are used to aide in determining divisibility, finding greatest common factors, least common multiples, and common denominators. Through experimentation, classroom examples, and homework, the author has introduced students to prime numbers and their applications.

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Chan, Ching-yin, and 陳靖然. "On k-tuples of almost primes." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2013. http://hdl.handle.net/10722/195967.

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Alazmi, Amal Abdullah. "The Prime Number Theorem: Elementary Results." Kent State University / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=kent1564916947385846.

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Alexander, Anita Nicole. "A HISTORY OF THE PRIME NUMBER THEOREM." Kent State University / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=kent1416827548.

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Alghamdi, Maha Mosaad. "ANALYTIC PROOF OF THE PRIME NUMBER THEOREM." Kent State University / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=kent1550224160190008.

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9

Shahabi, Majid. "The distribution of the classical error terms of prime number theory." Thesis, Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science, c2012, 2012. http://hdl.handle.net/10133/3252.

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Maynard, James. "Topics in analytic number theory." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:3bf4346a-3efe-422a-b9b7-543acd529269.

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In this thesis we prove several different results about the number of primes represented by linear functions. The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(phi(q)log{x}) for some value C depending on log{x}/log{q}. Different authors have provided different estimates for C in different ranges for log{x}/log{q}, all of which give C>2 when log{x}/log{q} is bounded. We show in Chapter 2 that one can take C=2 provided that log{x}/log{q}> 8 and q is sufficiently large. Moreover, we also produce a lower bound of size x/(q^{1/2}phi(q)) when log{x}/log{q}>8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem. Let k>1 and Pi(n) be the product of k linear functions of the form a_in+b_i for some integers a_i, b_i. Suppose that Pi(n) has no fixed prime divisors. Weighted sieves have shown that for infinitely many integers n, the number of prime factors of Pi(n) is at most r_k, for some integer r_k depending only on k. In Chapter 3 and Chapter 4 we introduce two new weighted sieves to improve the possible values of r_k when k>2. In Chapter 5 we demonstrate a limitation of the current weighted sieves which prevents us proving a bound better than r_k=(1+o(1))klog{k} for large k. Zhang has shown that there are infinitely many intervals of bounded length containing two primes, but the problem of bounded length intervals containing three primes appears out of reach. In Chapter 6 we show that there are infinitely many intervals of bounded length containing two primes and a number with at most 31 prime factors. Moreover, if numbers with up to 4 prime factors have `level of distribution' 0.99, there are infinitely many integers n such that the interval [n,n+90] contains 2 primes and an almost-prime with at most 4 prime factors.

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Kong, Yafang, and 孔亚方. "On linear equations in primes and powers of two." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2013. http://hub.hku.hk/bib/B50533769.

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It is known that the binary Goldbach problem is one of the open problems on linear equations in primes, and it has the Goldbach-Linnik problem, that is, representation of an even integer in the form of two odd primes and powers of two, as its approximate problem. The theme of my research is on linear equations in primes and powers of two. Precisely, there are two cases: one pair of linear equations in primes and powers of two, and one class of pairs of linear equations in primes and powers of two, in this thesis. In 2002, D.R. Heath-Brown and P.C. Puchta obtained that every sufficiently large even integer is the sum of two odd primes and k powers of two. Here k = 13, or = 7 under the generalized Riemann hypothesis. In 2010, B. Green and T. Tao obtained that every pair of linear equations in four prime variables with coefficients matrix A = (a_ij)s×t with s ≤ t, satisfying nondegenerate condition, that is, A has full rank and the only elements of the row-space of A over Q with two or fewer nonzero entries is the zero vector, is solvable. The restriction on the coefficient matrix means that they excluded the case of the binary Goldbach problem. Motivated by the above results, it is obtained that for every pair of sufficiently large positive even integers B1, B2, the simultaneous equation {█({B1 = p1 + p2 + 2v1 + 2v2 + · · · + 2vk ,@B2 = p3 + p4 + 2v1 + 2v2 + · · · + 2vk ,)┤ (1) is solvable, where p1, · · · , p4 are odd primes, each vi is a positive integer, and the positive integer k ≥ 63 or ≥ 31 under the generalized Riemann hypothesis. Note that, in 1989, M.C. Liu and K.M. Tsang have obtained that subject to some natural conditions on the coefficients, every pair of linear equations in five prime variables is solvable. Therefore one class of pairs of linear equations in four prime variables with special coefficient matrix and powers of two is considered. Indeed, it is deduced that every pair of integers B1 and B2 satisfying B1 ≡ 0 (mod 2), 3BB1 > e^(eB^48 ), B2 ≡ ∑_1^4▒= 1^(a_i ) (mod 2) and |B2| < BB1, where B = max1≤j≤4(2, |aj|), can be represented as {█(B1 = 〖p1〗_1 + p2 + 2^(v_1 ) + 2^(v_2 )+ · · · + 2^(v_k )@B2 = a1p1 + a2p2 + a3p3 + a4p4 + 2^(v_1 )+ 2^(v_2 )+ · · · + 2^(v_k ) )┤ (2) with k being a positive integer. Here p1, · · · p4 are odd primes, each 〖v 〗_iis a positive integer and the integral coefficients ai (i = 1, 2, 3, 4) satisfy {█((〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) = 1,@〖a 〗_1 〖a 〗_2< 0, 〖a 〗_3 〖a 〗_4<0,)┤ Moreover it is calculated that the positive integer k ≥ g(〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) where g(〖a 〗_21- 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = [(log⁡〖G(〖a 〗_21, …, 〖a 〗_24 〗)-log⁡〖F (〖a 〗_21, …, 〖a 〗_24)〗)/log0.975805-84.0285], (3) G(〖a 〗_21, 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = (min(1/(|a_24 |), 1/(|a_23 |)) - (〖|a〗_(21 )- a_22 |)/(|〖a_23 a〗_24 |) 〖(3B)〗^(-1) ×〖(3B)〗^(-1) (1-0.000001)- 〖(3B)〗^(-1-4), with B = max1≤j≤4(2, |a2j|), and F(a_21, …, a_24) = √(f(a_21)f〖(a〗_22 )) with f(a_2i) = {█(4414.15h (a_21-1)+5.088331 if a_21≠1@59.8411 if a_21=1,)┤ for i = 1, 2, and h(n) =∏_(p|n,p>2)▒(p-1)/(p-2). This result, if without the powers of two, can make up some of the cases excluded in Green and Tao’s paper.
published_or_final_version
Mathematics
Doctoral
Doctor of Philosophy

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Vlasic, Andrew. "A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions." Thesis, University of North Texas, 2004. https://digital.library.unt.edu/ark:/67531/metadc4476/.

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We follow a research paper that J. Elstrodt published in 1998 to prove the Prime Number Theorem for arithmetic progressions. We will review basic results from Dirichlet characters and L-functions. Furthermore, we establish a weak version of the Wiener-Ikehara Tauberian Theorem, which is an essential tool for the proof of our main result.

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Siu, Wai-chuen. "Small prime solutions of some ternary equations /." Hong Kong : University of Hong Kong, 1995. http://sunzi.lib.hku.hk/hkuto/record.jsp?B17538191.

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Bronder, Justin S. "The AKS Class of Primality Tests: A Proof of Correctness and Parallel Implementation." Fogler Library, University of Maine, 2006. http://www.library.umaine.edu/theses/pdf/BronderJS2006.pdf.

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15

Mullen, Woodford Roger. "Partitions into prime powers and related divisor functions." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/1246.

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In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

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Dudek, Adrian W. "3MT : A fine time to find primes." Thesis, https://www.youtube.com/watch?v=g0qbNksZLgo&list=PL8rZPGPMzfuK7yVuY31rWGFkHM_DF1ItU&index=4, 2013. http://hdl.handle.net/1885/13619.

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We all have a shared history; when we were in primary school, our teachers told us that a number is prime if it’s only divisible by one and itself. We might also share severe scarring, from when we popped our little hand in the air and asked the question: primes - what are they good for?

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17

Wilson, Keith Eirik. "Factoring Semiprimes Using PG2N Prime Graph Multiagent Search." PDXScholar, 2011. https://pdxscholar.library.pdx.edu/open_access_etds/219.

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In this thesis a heuristic method for factoring semiprimes by multiagent depth-limited search of PG2N graphs is presented. An analysis of PG2N graph connectivity is used to generate heuristics for multiagent search. Further analysis is presented including the requirements on choosing prime numbers to generate 'hard' semiprimes; the lack of connectivity in PG1N graphs; the counts of spanning trees in PG2N graphs; the upper bound of a PG2N graph diameter and a conjecture on the frequency distribution of prime numbers on Hamming distance. We further demonstrated the feasibility of the HD2 breadth first search of PG2N graphs for factoring small semiprimes. We presented the performance of different multiagent search heuristics in PG2N graphs showing that the heuristic of most connected seedpick outperforms least connected or random connected seedpick heuristics on small PG2N graphs of size N

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蕭偉泉 and Wai-chuen Siu. "Small prime solutions of some ternary equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1995. http://hub.hku.hk/bib/B31213595.

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Domingues, Riaal. "A polynomial time algorithm for prime recognition." Diss., Pretoria : [s.n.], 2006. http://upetd.up.ac.za/thesis/available/etd-08212007-100529.

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BATTISTONI, FRANCESCO. "APPLICATIONS OF PRIME DENSITIES IN NUMBER THEORY AND CLASSIFICATION OF NUMBER FIELDS WITH BOUNDED INVARIANTS." Doctoral thesis, Università degli Studi di Milano, 2020. http://hdl.handle.net/2434/703505.

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This Ph.D. thesis collects the author's works and interests in several parts of Number Theory, from algebraic problems related to relations between number fields which are based on the factorization of prime numbers in the rings of integers, up to the application of tools concerning the density of primes with given splitting type in number fields to the computation of the average rank of specific families of elliptic curves, concluding finally with the classification and estimate of the main invariants of a number field, like the discriminant and the regulator, pursued by means of analytic formulas and algorithmic methods developed on the previous tools and implemented on suitable computer algebra systems, like PARI/GP.

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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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Suresh, Arvind. "On the Characterization of Prime Sets of Polynomials by Congruence Conditions." Scholarship @ Claremont, 2015. http://scholarship.claremont.edu/cmc_theses/993.

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This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.

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Henderson, Cory. "Exploring the Riemann Hypothesis." Kent State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=kent1371747196.

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Fraser, Robert. "On the number of prime solutions to a system of quadratic equations." Thesis, University of British Columbia, 2013. http://hdl.handle.net/2429/44283.

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Consider the system of quadratic diophantine equations bX² − aY² = 0 bX • Y − eY² = 0 constrained to the prime numbers contained in the box [0,N]²ⁿ. The Hardy- Littlewood circle method is applied to show that, under some local conditions on a, b, and e, the number of prime solutions contained in the box is asymptotic to a constant times N²ⁿ⁻⁴/ (logN)²ⁿ , where the constant depends on a, b, and e.

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Kawaguchi, Yuki. "Near Miss abc-Triples in General Number Fields." Kyoto University, 2019. http://hdl.handle.net/2433/242576.

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Montemayor, Anthony. "On Nullification of Knots and Links." TopSCHOLAR®, 2012. http://digitalcommons.wku.edu/theses/1158.

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Motivated by the action of XER site-specific recombinase on DNA, this thesis will study the topological properties of a type of local crossing change on oriented knots and links called nullification. One can define a distance between types of knots and links based on the minimum number of nullification moves necessary to change one to the other. Nullification distances form a class of isotopy invariants for oriented knots and links which may help inform potential reaction pathways for enzyme action on DNA. The minimal number of nullification moves to reach a è-component unlink will be called the è-nullification number. This thesis will demonstrate the relationship of the nullification numbers to a variety of knot invariants, and use these to solve the è-nullification numbers for prime knots up to 10 crossings for any è. A table of nullification numbers for oriented prime links up to 9 crossings is also presented, but not all cases are solved. In addition, we examine the families of rational links and torus links for explicit results on nullification. Nullification numbers of torus knots and a subfamily of rational links are solved. In doing so, we obtain an expression for the four genus of said subfamily of rational links, and an expression for the nullity of any torus link.

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Connors, Richard D. "Classical periodic orbit correlations and quantum spectral statistics." Thesis, University of Bristol, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.246244.

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Streipel, Jakob. "On the Number of Periodic Points of Quadratic Dynamical Systems Modulo a Prime." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-45635.

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We investigate the number of periodic points of certain discrete quadratic maps modulo prime numbers. We do so by first exploring previously known results for two particular quadratic maps, after which we explain why the methods used in these two cases are hard to adapt to a more general case. We then perform experiments and find striking patterns in the behaviour of these general cases which suggest that, apart from the two special cases, the number of periodic points of all quadratic maps of this type behave the same. Finally we formulate a conjecture to this effect.

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Jacobsson, Mattias. "Bitwise relations between n and φ(n) : A novel approach at prime number factorization." Thesis, Blekinge Tekniska Högskola, Institutionen för datalogi och datorsystemteknik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:bth-16655.

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Cryptography plays a crucial role in today’s society. Given the influence, cryptographic algorithms need to be trustworthy. Cryptographic algorithms such as RSA relies on the problem of prime number factorization to provide its confidentiality. Hence finding a way to make it computationally feasible to find the prime factors of any integer would break RSA’s confidentiality. The approach presented in this thesis explores the possibility of trying to construct φ(n) from n. This enables factorization of n into its two prime numbers p and q through the method presented in the original RSA paper. The construction of φ(n) from n is achieved by analyzing bitwise relations between the two. While there are some limitations on p and q this thesis can in favorable circumstances construct about half of the bits in φ(n) from n. Moreover, based on the research a conjecture has been proposed which outlines further characteristics between n and φ(n).

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Segarra, Elan. "An Exploration of Riemann's Zeta Function and Its Application to the Theory of Prime Distribution." Scholarship @ Claremont, 2006. https://scholarship.claremont.edu/hmc_theses/189.

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Identified as one of the 7 Millennium Problems, the Riemann zeta hypothesis has successfully evaded mathematicians for over 100 years. Simply stated, Riemann conjectured that all of the nontrivial zeroes of his zeta function have real part equal to 1/2. This thesis attempts to explore the theory behind Riemann’s zeta function by first starting with Euler’s zeta series and building up to Riemann’s function. Along the way we will develop the math required to handle this theory in hopes that by the end the reader will have immersed themselves enough to pursue their own exploration and research into this fascinating subject.

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Padilha, José Cleiton Rodrigues. "Números primos." Universidade Federal da Paraíba, 2013. http://tede.biblioteca.ufpb.br:8080/handle/tede/7534.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The purpose of this work is to present a special category of integers: Prime numbers. It will be presented a historical retrospective, quoting the most important and interesting results achieved by great mathematicians over the years. Then, most of these results will be formally announced with propositions or theorems and their respective demonstrations, starting with the basic properties of divisibility and cul- minating in some primality tests.
O propósito deste trabalho é apresentar uma categoria especial de números inteiros: Os Números Primos. Será apresentada uma retrospectiva histórica,citando os resultados mai s importantes e interessantes obtidos por grandes matemáticos ao longodos anos. Em seguida, a maioria destes resultados serão formalmente enunciados com proposições ou teoremas e suas respectivas demonstrações,começando com as propriedades básicas da divisibilidade e culminando em alguns testes de primalidade.

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Rezola, Nolberto. "Unique Prime Factorization of Ideals in the Ring of Algebraic Integers of an Imaginary Quadratic Number Field." CSUSB ScholarWorks, 2015. https://scholarworks.lib.csusb.edu/etd/205.

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The ring of integers is a very interesting ring, it has the amazing property that each of its elements may be expressed uniquely, up to order, as a product of prime elements. Unfortunately, not every ring possesses this property for its elements. The work of mathematicians like Kummer and Dedekind lead to the study of a special type of ring, which we now call a Dedekind domain, where even though unique prime factorization of elements may fail, the ideals of a Dedekind domain still enjoy the property of unique prime factorization into a product of prime ideals, up to order of the factors. This thesis seeks to establish the unique prime ideal factorization of ideals in a special type of Dedekind domain: the ring of algebraic integers of an imaginary quadratic number field.

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Green, Nathan Eric. "Integral Traces of Weak Maass Forms of Genus Zero Odd Prime Level." BYU ScholarsArchive, 2013. https://scholarsarchive.byu.edu/etd/4161.

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Duke and Jenkins defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.

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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.

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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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Richardson, Robert. "On the Number of Integers Expressible as the Sum of Two Squares." Connect to resource online, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1265123768.

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McCulloch, Catherine Margaret. "Discrete logarithm problem over finite prime fields." Thesis, Queensland University of Technology, 1998. https://eprints.qut.edu.au/36976/1/36976_McCulloch_1988.pdf.

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Difficulty in solving the discrete logarithm problem has led to its use in key exchange, public key cryptography and digital signatures. To measure the security of these algorithms, it is necessary to evaluate the methods currently available for attack. Although the applications of the discrete logarithm problem can be implemented in a variety of different groups, only implementations over multiplicative integers modulo a large prime p are considered. The object of this work is to review the current methods of solving the discrete logarithm problem - key exhaustion, Shanks' baby-step giant-step algorithm, Pollard's rho algorithm, the Silver Pohlig Hellman algorithm, index calculus methods and the general number field sieve. The resulting document contains all relevant mathematics, theorems and algorithms. As only one workstation is used, the problem will not be solved for large primes, but an indication of the relative strengths and weaknesses of each algorithm will be gained. Both the theoretical and practical issues were considered when comparing the attacks available. The algorithms were implemented using the computer algebra system "Magma", which was developed at the University of Sydney. Magma was chosen as it is a flexible package that is not restricted to group theory. The source code is included in Appendix B. The simplest methods to implement are key exhaustion, which relies on testing all possibilities, and the first improvement on this method - Shank's baby-step giant-step algorithm. Both methods are infeasible when the prime number is large. Pollard's rho algorithm, again impractical for large p, has the same expected running time as Shank's baby-step giant-step algorithm, but the storage requirements are negligible. The Silver Pohlig Hellman algorithm which is again impractical for a large p unless p-1 has small factors is also covered. Index calculus methods offer improvements in the time involved to attack the system, once the prime number becomes too large for the earlier methods. Unlike the previous algorithms, the index calculus methods are not generic, they can only be used for particular groups, one of which is the field GF(p ), considered here. The methods involve two parts, a costly precomputation stage that needs to be performed only once for each prime, and the calculation of the individual logarithm. Three methods are investigated - the first by McCurley, the second by Coppersmith, Odlyzko and Schroeppel, and the third by LaMacchia and Odlyzko. By attacking with these methods, primes with fewer than 200 bits are insecure and primes with less than 512 bits should be avoided. By adapting the general number field sieve to solve logarithms, the running time of the attack in some instances can be further improved. Unlike the index calculus methods, the time required for the precomputation and that required for the evaluation of the individual logarithm, are similar. This perhaps reduces the usefulness of the algorithm in the case where the same attack is to be implemented a number of times to determine several different logarithms. If the same prime is to be used for a number of attacks, it may be quicker to use an index calculus method as the precomputation is performed once and then the logarithms can be quickly calculated.

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37

Clark, John. "On a conjecture involving Fermat's Little Theorem." [Tampa, Fla] : University of South Florida, 2008. http://purl.fcla.edu/usf/dc/et/SFE0002485.

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38

Tramte, Daniel A. "Alter-Soni-Cation." Bowling Green State University / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1272646252.

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39

Nilsson, Marcus. "Monomial Dynamical Systems in the Fields of p-adic Numbers and Their Finite Extensions." Doctoral thesis, Växjö universitet, Matematiska och systemtekniska institutionen, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:vxu:diva-403.

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40

Månsson, Jakob. "Comparative Study of CPU and GPGPU Implementations of the Sievesof Eratosthenes, Sundaram and Atkin." Thesis, Blekinge Tekniska Högskola, Institutionen för datavetenskap, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:bth-21111.

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Background. Prime numbers are integers divisible only on 1 and themselves, and one of the oldest methods of finding them is through a process known as sieving. A prime number sieving algorithm produces every prime number in a span, usually from the number 2 up to a given number n. In this thesis, we will cover the three sieves of Eratosthenes, Sundaram, and Atkin. Objectives. We shall compare their sequential CPU implementations to their parallel GPGPU (General Purpose Graphics Processing Unit) counterparts on the matter of performance, accuracy, and suitability. GPGPU is a method in which one utilizes hardware indented for graphics rendering to achieve a high degree of parallelism. Our goal is to establish if GPGPU sieving can be more effective than the sequential way, which is currently commonplace. Method. We utilize the C++ and CUDA programming languages to implement the algorithms, and then extract data regarding their execution time and accuracy. Experiments are set up and run at several sieving limits, with the upper bound set by the memory capacity of available GPU hardware. Furthermore, we study each sieve to identify what characteristics make them fit or unfit for a GPGPU approach. Results. Our results show that the sieve of Eratosthenes is slow and ill-suited for GPGPU computing, that the sieve of Sundaram is both efficient and fit for parallelization, and that the sieve of Atkin is the fastest but suffers from imperfect accuracy. Conclusions. Finally, we address how the lesser concurrent memory capacity available for GPGPU limits the ranges that can be sieved, as compared to CPU. Utilizing the beneficial characteristics of the sieve of Sundaram, we propose a batch-divided implementation that would allow the GPGPU sieve to cover an equal range of numbers as any of the CPU variants.

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41

Klembalski, Katharina. "Cryptography and number theory in the classroom -- Contribution of cryptography to mathematics teaching." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-80390.

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Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts.

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42

Johansson, Angela. "Distributed System for Factorisation of Large Numbers." Thesis, Linköping University, Department of Electrical Engineering, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-1883.

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This thesis aims at implementing methods for factorisation of large numbers. Seeing that there is no deterministic algorithm for finding the prime factors of a given number, the task proves rather difficult. Luckily, there have been developed some effective probabilistic methods since the invention of the computer so that it is now possible to factor numbers having about 200 decimal digits. This however consumes a large amount of resources and therefore, virtually all new factorisations are achieved using the combined power of many computers in a distributed system.

The nature of the distributed system can vary. The original goal of the thesis was to develop a client/server system that allows clients to carry out a portion of the overall computations and submit the result to the server.

Methods for factorisation discussed for implementation in the thesis are: the quadratic sieve, the number field sieve and the elliptic curve method. Actually implemented was only a variant of the quadratic sieve: the multiple polynomial quadratic sieve (MPQS).

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43

Vychodil, Petr. "Softwarová podpora výuky kryptosystémů založených na problému faktorizace velkých čísel." Master's thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2009. http://www.nusl.cz/ntk/nusl-218146.

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This thesis deals with new teaching software, which supports asymmetric encryption algorithms based on the issue of large numbers´ factorization. A model program was created. It allows to carry out operations related to encryption and decryption with an interactive control. There is a simple way to understand the principle of the RSA encryption method with its help. The encryption of algorithms is generally analysed in chapters 1,2. Chapters 3 and 4 deals with RSA encryption algorithm in much more details, and it also describes the principles of the acquisition, management and usage of encryption keys. Chapters 5 suggest choosing of appropriate technologies for the creation of the final software product, which allow an appropriate way to present the characteristics of the extended RSA encryption algorithm. The final software product is the java applet. Aplet is described in the chaprers 6 and 7. It is divided into a theoretical and practical part. The theoretical part presents general information about the RSA encryption algorithm. The practical part allows the users of the program to have a try at tasks connected with the RSA algorthm in their own computers. The last part of Java applet deals with the users´ work results. The information obtained by the user in the various sections of the program are satisfactory enough to understand the principle of this algorithm´s operations.

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44

Chellali, Mustapha. "Congruences, nombres de Bernoulli et polynômes de Bessel." Université Joseph Fourier (Grenoble ; 1971-2015), 1989. http://www.theses.fr/1989GRE10091.

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En premiere partie, on donne des congruences entre nombres de bernoulli-hcowitz dans le cas supersingulier. En deuxieme partie, on montre que la suite des nombres de bernoulli verifie des formules de recurrence qui servent a tester si un nombre premier est irregulier. En troisieme partie, on etudie les zeros des polynomes de bessel generalises, en particulier on encadre un zero reel, apres developpement asymptotique, et on donne des estimations uniformes des valeurs de ces polynomes

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Rigoti, Marcio Dominicali. "Números primos: os átomos dos números." Universidade Tecnológica Federal do Paraná, 2014. http://repositorio.utfpr.edu.br/jspui/handle/1/1075.

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CAPES
Este trabalho apresenta um estudo sobre os Números Primos que passa por resultados básicos, como a infinitude dos números primos e o Teorema Fundamental da Aritmética, e resultados mais sofisticados, como o Teorema de Wilson e a consequente função geradora de primos. Além dos resultados teóricos apresenta-se uma interpretação geométrica para os números primos. Essa interpretação e aplicada na ilustração de alguns dos resultados relacionados a primos abordados no ensino básico. Atividades envolvendo a interpretação geométrica apresentada são sugeridas no capítulo final.
This work presents a study about Prime Numbers, since basic results, like the prime number’s infinity and the Arithmetic Fundamental Theorem, to more sophisticated results, as Wilson’s Theorem and it’s consequent Prime generating function. Further the theoretical results we present a prime’s geometric interpretation. This interpretation is applied to illustrate some results related to primes, which appears in basic education. Activities about this geometric interpretation are suggested in the final chapter.

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46

Anicama, Jorge. "Prime numbers and encryption." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/95565.

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In this article we will deal with the prime numbers and its current use in encryption algorithms. Encryption algorithms make possible the exchange of sensible data in internet, such as bank transactions, email correspondence and other internet transactions where privacy is important.

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47

Goldoni, Luca. "Prime Numbers and Polynomials." Doctoral thesis, Università degli studi di Trento, 2010. https://hdl.handle.net/11572/368684.

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This thesis deals with the classical problem of prime numbers represented by polynomials. It consists of three parts. In the first part I collected many results about the problem. Some of them are quite recent and this part can be considered as a survey of the state of the art of the subject. In the second part I present two results due to P. Pleasants about the cubic polynomials with integer coefficients in several variables. The aim of this part is to simplify the works of Pleasants and modernize the notation employed. In such a way these important theorems are now in a more readable form. In the third part I present some original results related with some algebraic invariants which are the key-tools in the works of Pleasants. The hidden diophantine nature of these invariants makes them very difficult to study. Anyway some results are proved. These results make the results of Pleasants somewhat more effective.

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48

Goldoni, Luca. "Prime Numbers and Polynomials." Doctoral thesis, University of Trento, 2010. http://eprints-phd.biblio.unitn.it/384/1/Thesis.pdf.

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This thesis deals with the classical problem of prime numbers represented by polynomials. It consists of three parts. In the first part I collected many results about the problem. Some of them are quite recent and this part can be considered as a survey of the state of the art of the subject. In the second part I present two results due to P. Pleasants about the cubic polynomials with integer coefficients in several variables. The aim of this part is to simplify the works of Pleasants and modernize the notation employed. In such a way these important theorems are now in a more readable form. In the third part I present some original results related with some algebraic invariants which are the key-tools in the works of Pleasants. The hidden diophantine nature of these invariants makes them very difficult to study. Anyway some results are proved. These results make the results of Pleasants somewhat more effective.

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49

Wolczuk, Dan. "Intervals with few Prime Numbers." Thesis, University of Waterloo, 2004. http://hdl.handle.net/10012/1064.

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In this thesis we discuss some of the tools used in the study of the number of primes in short intervals. In particular, we discuss a large sieve density estimate due to Gallagher and two classical delay equations. We also show how these tools have been used by Maier and Stewart and provide computational data to their result.

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50

Shiu, Daniel Kai Lun. "Prime numbers in arithmetic progressions." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.318815.

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Dissertations / Theses on the topic 'Prime number'

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