Dissertations / Theses on the topic 'Mersenne'

Cette thèse doctorale prétend contribuer, premièrement, à soumettre à discussioncette interprétation dominante de la pensée philosophique et scientifique du PèreMinime Marin Mersenne (1588-1648), étant portée et structurée par la question duscepticisme, et deuxièmement, à mettre en avant la manière dont cette philosophieincarne l’esprit de la révolution scientifique du XVIIe siècle par sa capacité de se mettreen question dans sa recherche insatiable de la vérité; une recherche accompagnée d’unsouci de conservation de l’ordre politique et religieux. L’hypothèse principale de notretravail consiste à affirmer une profonde transformation dans la conception de la musique: transformation qui mène d’une science quadriviale et subalterne aux mathématiques,exigeant la soumission au jugement de la raison dans la pratique, à une science physiqueet mathématique, dont la recherche se fonde sur de nombreuses expériences,reconnaissant l’individualité de l’expérience esthétique, la liberté de l’imagination descompositeurs et le caractère ineffable du sublime musical. Il s’agit d’une transformationqui n’est pas exempte de difficultés, car elle ne conduit pas simplement à affirmerl’existence de deux périodes dans la pensée de Mersenne. En effet, il exprime ses doutessur la pertinence de l’approche spéculative lorsqu’il discute avec ses correspondants surla réforme musicale proposée à l’imitation des anciens et ne cesse de rappeler laperfection des rapports numériques des consonances lorsqu’il est prêt à les mettre enquestion en acceptant et en cherchant les fondements de la pratique de l’accordage desinstruments. Or, malgré cette complexité (voire ces contradictions) nous jugeons et nousprétendons montrer que cette transformation est indéniable et que l’épistémologie duMinime doit être analysée à la lumière des problématiques et de nouvelles expériencesscientifiques auxquelles il est confronté et non comme une manière de donner réponseaux arguments du scepticisme
This PhD dissertation provides a critical perspective of the dominantinterpretation of the scientific and philosophical works of Father Marin Mersenne(1588-1648) entirely structured by the sceptical question. The development of his ideasabout music embodies the spirit of the scientific revolution which emerges in theseventeenth-century. His investigation has the capacity to put his methods into questionwith an insatiable quest for the truth; a quest that involves political and religiousconcerns. The aim of this study is to show a profound transformation in the conceptionof music. This transformation that leads from a science of the quadrivium (subordinateto mathematics and claiming superiority of the judgement of reason) to a physical andmathematical science grounded on experience that recognizes the individuality of theesthetic experience, the liberty of the imagination of the composers and the ineffabilityof the sublime. It is quite difficult however to identify the existence of two differentstages in Mersenne’s thought. It is surprising how he expresses doubts about therelevance of the speculative approach to music whereas a musical reform is proposed inhis apologetic writings, having as a model the perfection of proportions of consonancesand rhythmic combinations well known by the ancients. And also, when he accepts thepractice of musical temperament, challenging the observation of the mathematicalperfection, he will continue to remind the proportions underlying the consonances.Despite this complexity (even these contradictions) we consider and pretend to showthat this transformation is undeniable and that the Mersenne epistemology must beanalysed according to the scientific questions and experiences which he faces in hisinvestigations and not as a response to a sceptical crisis

Encryption is very much a vast subject covering myriad techniques to conceal and safeguard data and communications. Of the techniques that are available, methodologies that incorporate the number theoretic transforms (NTTs) have gained recognition, specifically the new Mersenne number transform (NMNT). Recently, two new transforms have been introduced that extend the NMNT to a new generalised suite of transforms referred to as the generalised NMNT (GNMNT). These two new transforms are termed the odd NMNT (ONMNT) and the odd-squared NMNT (O2NMNT). Being based on the Mersenne numbers, the GNMNTs are extremely versatile with respect to vector lengths. The GNMNTs are also capable of being implemented using fast algorithms, employing multiple and combinational radices over one or more dimensions. Algorithms for both the decimation-in-time (DIT) and -frequency (DIF) methodologies using radix-2, radix-4 and split-radix are presented, including their respective complexity and performance analyses. Whilst the original NMNT has seen a significant amount of research applied to it with respect to encryption, the ONMNT and O2NMNT can utilise similar techniques that are proven to show stronger characteristics when measured using established methodologies defining diffusion. Analyses in diffusion using a small but reasonably sized vector-space with the GNMNTs will be exhaustively assessed and a comparison with the Rijndael cipher, the current advanced encryption standard (AES) algorithm, will be presented that will confirm strong diffusion characteristics. Implementation techniques using general-purpose computing on graphics processing units (GPGPU) have been applied, which are further assessed and discussed. Focus is drawn upon the future of cryptography and in particular cryptology, as a consequence of the emergence and rapid progress of GPGPU and consumer based parallel processing.

La matière de cette thèse vient de 261 écrits sur la musique décrits (2d vol. ) dans un catalogue systématique (notices et index). Le 1er vol. Est nécessaire à l'étude des aspects fondamentaux du corpus : les modèles de la théorie musicale. Le discours sur la musique est étudié du point de vue de la distance qu'il prend de l'héritage du passé (antique, médiéval, humaniste) : de l'appropriation de cet héritage et de ses avatars. La multitude des modèles potentiels, de leurs interprétations, puis des finalités et des approches des auteurs, génére plusieurs confrontations : Anciens/modernes, théoriciens/praticiens, Français/italiens. Ces visions bipolaires de la musique témoignent de l'impossibilité de la circonscrire par un discours propre. Sont ensuite examinés les liens entre la musique et la science, notamment les aspects de sa codification théorique, telles la compréhension spectrale du son, l'étude des tempéraments musicaux, les recherches d'étalons du temps et de la hauteur.

Le manuscrit 2884 est un autographe du père Mersenne, unique exemplaire conservé à la bibliothèque de l’Arsenal. Cet écrit constitué en livre et chapitres est une théorie spéculative du son et des consonances inachevée. Comme le manuscrit n’a jamais été édité nous en proposons la transcription selon deux pratiques différentes, une selon le modèle de l’École nationale des Chartes et l’autre diplomatique selon le laboratoire de l’ITEM spécialisé dans les brouillons d’écriture. La transcription est complétée par des notes critiques et historiques, une table des théorèmes reconstituée, d’un index des noms propres. Le fac-simile du manuscrit est proposé dans un volume à part. Dans la première partie une étude codicologique permet de situer le manuscrit dans son aspect matériel. Une étude génétique détermine ses liens avec les imprimés du père Mersenne et retrace son parcours général de rédaction. La dernière partie de la thèse tente de cerner les motifs de son abandon par une analyse comparative avec quelques livres de l’Harmonie Universelle. Notre hypothèse de départ repose sur un changement de philosophie naturelle, passage de la philosophie aristotélicienne à la philosophie d’inspiration galiléenne. En effet nous montrons que tout le discours tenu dans le manuscrit 2884 est ontologiquement fondé dans la conception aristotélicienne alors que les mêmes sujets abordés dans l’Harmonie Universelle manifestent une rupture avec ce fondement ontologique ainsi qu’une profonde évolution de ses idées
Manuscript 2884 is an autograph by Marin Mersenne, a unique document preserved in the Arsenal Library. This writing formed of books and chapters is an unfinished speculative theory of sound and consonance. As the manuscript has never been edited we propose a transcription following two different usages, the first according to the model of the École nationale des Chartes and the second a diplomatic transcription according to the laboratory of the ITEM (Institut des Textes et Manuscrits modernes) specializing in drafts of texts. The transcriptions are supplemented by critical and historical notes, a reconstructed table of theorems and an index of proper names. The facsimile of the manuscript is proposed in a separate volume. In the first part, a codicological study makes it possible to describe the manuscript in its material aspect. A genetic study determines its links with the printed works of Marin Mersenne and traces its general course of writing. The last part of the thesis attempts to define the reasons for its abandonment by a comparative analysis with some books of Harmonie Universelle. Our initial hypothesis supposes a change in natural philosophy, a change from Aristotelian philosophy to a philosophy of Galilean inspiration. We show that the discourse held in manuscript 2884 as a whole is ontologically based in the Aristotelian conception whereas the same subjects approached in Harmonie Universelle show a break with this ontological foundation as well as a profound evolution of Mersenne’s ideas

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This work presents a study of prime numbers, how they are distributed, how many prime numbers are there between 1 and a real number x, formulas that generate primes, and a generalization to Bertrand's Postulate. Six proofs that there are in nitely many primes using reductio ad absurdum, Fermat numbers, Mersenne numbers, Elementary Calculus and Topology are discussed.
Este trabalho apresenta um estudo sobre os números primos, como estão distribu ídos, quantos números primos existem entre 1 e um número real x qualquer, fórmulas que geram primos, além de uma generalização para o Postulado de Bertrand. São abordadas seis demonstrações que mostram que existem in nitos números primos usando redução ao absurdo, Números de Fermat, Números de Mersenne, Cálculo Elementar e Topologia.

Thesis (Ph. D.)--Indiana University, 2005.
Computer printout. Source: Dissertation Abstracts International, Volume: 66-02, Section: A, page: 0404. Chair: Frank Samarotto. Includes bibliographical references (leaves 407-419), abstract, and vita.

S'han analitzat els monocordis de Ramos de Pareja, Zarlino i Mersenne, a partir dels textos i imatges que els descriuen, s'han reconstruït com a objectes per a l'experimentació i s'han contextualitzat en la historiografia dels instruments científics de l'anomenada "Revolució Científica ". Associats al concepte "consonància", cada monocordi salvaguardava una "veritat" físic-matemàtica (harmònics naturals, llindars psicoacústics, xarxa de variables físiques) i, alhora, posseïa funcions pràctiques com a eina per a la pedagogia, construcció i afinació d'instruments musicals . Aquesta doble funció (científica i estètica), era indissoluble, i per això, les classificacions historiogràfiques que insisteixen a ordenar els instruments científics segons la seva naturalesa filosòfica o pràctica, no poden aplicar-se al monocordi, deixant de costat. Si durant els segles XV, XVI i XVII la "música" va ser una "ciència", el monocordi és un instrument científic, i com a tal, sorprèn la seva absència i es reclama i reivindica la seva presència a les col•leccions i museus d'instruments musicals i científics . Les reconstruccions aquí presentades-reals i virtuals-ajuden a comprendre un saber en què confluïen sensibilitat i raó però que, finalment, va derivar cap a un exclusiu interès formal i quantitatiu -la naixent ciència moderna- que podia prescindir de tota estètica a priori, com fonament de la seva recerca.
Se han analizado los monocordios de Ramos de Pareja, Zarlino y Mersenne, a partir de los textos e imágenes que los describen; se han reconstruido como objetos para la experimentación y se han contextualizado en la historiografía de los instrumentos científicos de la llamada “Revolución Científica”. Asociados al concepto “consonancia”, cada monocordio salvaguardaba una “verdad” físico-matemática (armónicos naturales, umbrales psicoacústicos, red de variables físicas) y, a la vez, poseía funciones prácticas como herramienta para la pedagogía, construcción y afinación de instrumentos musicales. Esta doble función (científica y estética), era indisoluble, y por ello, las clasificaciones historiográficas que insisten en ordenar los instrumentos científicos según su naturaleza filosófica o práctica, no pueden aplicarse al monocordio, dejándolo de lado. Si durante los siglos XV, XVI y XVII la “música” fue una “ciencia”, el monocordio es un instrumento científico, y como tal, sorprende su ausencia y se reclama y reivindica su presencia en las colecciones y museos de instrumentos musicales y científicos. Las reconstrucciones aquí presentadas –reales y virtuales– ayudan a aprehender un saber en el que confluían sensibilidad y razón pero que, finalmente, derivó hacia un exclusivo interés formal y cuantitativo –la naciente ciencia moderna– que podía prescindir de toda estética a priori, como fundamento de su investigación.
The monochords described by Ramos de Pareja, Zarlino and Mersenne, have been analysed in their writings, reconstructed as experimental objects and contextualized in the historiography of the scientific instruments of the so-called "Scientific Revolution." Related to concept of “consonance”, each monochord preserved a physic-mathematical "truth" (natural harmonics, psychoacoustic thresholds, physical network variables) and, in turn, had practical functions as pedagogic tool, construction and tuning of musical instruments. This double function (scientific and aesthetic) was indissoluble, and therefore the historiographical classification that insists in ordering scientific instruments according to their philosophical or practical nature can not be applied to the monochord, leaving it aside. If "music" was a "science" itself in the 15th, 16th and 17th centuries, the monochord is a scientific instrument, and as such, its absence surprises and its presence is protested and vindicated of the collections and museums of musical and scientific instruments. The reconstructions presented –real and virtual– help to understand a knowledge where sense and reason blend, but which finally acquired an exclusive quantitative and formal interest –early modern science– that could dispensed an aesthetic a priori, as the basis of its research.

In this work, I focus on one of Galileo's concepts which was neither mathematically nor empirically derived, but instead based on a fundamental intuition regarding the nature of motion: that all mechanical phenomena can be treated in the same way, using the same mathematical and conceptual apparatus. This was Galileo's concept of 'correspondence', and I follow it from its origins at the turn of the 17th century through Thomas Harriot, Marin Mersenne and ultimately to Christiaan Huygens. At the centre of the concept of correspondence was that phenomena which looked similar really were the same; they were separate instances of the same fundamental processes. Hanging chains and projectile trajectories did not form the same curve by coincidence; they formed the same curve because both were produced by the same competition between vertical and horizontal tendencies. Correspondences were one of the major motivating and legitimising factors behind both Galileo and Huygens' desire to treat all of nature mathematically. This conceptual structure justified their treatment of all of mechanics as mathematically the same. Harriot and Mersenne's roles in this story are to show how contemporaries of Galileo could approach the same topic in drastically different ways. Unlike Huygens, neither Harriot nor Mersenne understood the concept of correspondences. While Galileo and Huygens relied crucially on correspondences to understand natural phenomena, both Harriot and Mersenne were able to achieve many important results in mechanics without it. This work is the biography of a concept; one that is contingent, constructed, frequently fruitful but not a historical or scientific necessity.

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In this thesis we study some topics of the Theory of Numbers as an inspiration for future studies of Perfect Numbers and Mersenne Primes. We present some important results for our study and analyze some statements of Fermat's Little Theorem, showing the various mathematical demonstrations that proved under various logical aspects. We have clari ed some historical aspects and conjectures for perfect numbers, through a simple narrative of facts and this will certainly give us the emphasis that have motivated and still motivates many mathematicians for the study of Perfect Numbers.
Nesta dissertação fazemos um estudo de alguns tópicos da Teoria dos Números como motivação para o estudo dos Números Perfeitos e Primos de Mersenne. Apresentamos alguns resultados importantes para o nosso estudo e analisamos algumas demonstrações do Pequeno Teorema de Fermat, evidenciando a demonstração de vários matemáticos que os provaram sob vários aspectos lógicos. Evidenciamos alguns aspectos históricos e conjecturas para os números perfeitos, através de uma narrativa simples dos fatos e que certamente nos dão a ênfase que motivou e motiva vários matemáticos para o estudo dos números perfeitos.

Dopo una breve introduzione storica ci si occupa del problema della dimostrazione della infinità dei numeri primi. Di questa si espongono cinque dimostrazioni diverse trovate nell'arco di più di duemila anni. La tesi è completata dall'esposizione di una serie di criteri di divisibilità utili nell'insegnamento primario e secondario completamente dimostrati.

A new class of moduli called the low-weight polynomial form integers (LWPFIs) is introduced. LWPFIs are expressed in a low-weight, monic polynomial form, p = f(t). While the generalized Mersenne numbers (GMNs) proposed by Solinas allow only powers of two for t, LWPFIs allow any positive integers. In our first proposal of LWPFIs, we limit the coefficients of f(t) to be 0 and ±1, but later we extend LWPFIs to allow any integer of less than t for the coefficients of f(t). Modular multiplication using LWPFIs is performed in two phases: 1) polynomial multiplication in Z[t]/f(t) and 2) coefficient reduction. We present an efficient coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm. We also show a coefficient reduction algorithm based on the Montgomery reduction algorithm. We give analysis and experimental results on modular multiplication using LWPFIs.

New three, four and five-way squaring formulae based on the Toom-Cook multiplication algorithm are presented. All previously known squaring algorithms are symmetric in the sense that the point-wise multiplication step involves only squarings. However, our squaring algorithms are asymmetric and use at least one multiplication in the point-wise multiplication step. Since squaring can be performed faster than multiplication, our asymmetric squaring algorithms are not expected to be faster than other symmetric squaring algorithms for large operand sizes. However, our algorithms have much less overhead and do not require any nontrivial divisions. Hence, for moderately small and medium size operands, our algorithms can potentially be faster than other squaring algorithms. Experimental results confirm that one of our three-way squaring algorithms outperforms the squaring function in GNU multiprecision library (GMP) v4. 2. 1 for certain range of input size. Moreover, for degree-two squaring in Z[x], our algorithms are much faster than any other squaring algorithms for small operands.

We present a side channel attack on XTR cryptosystems. We analyze the statistical behavior of simultaneous XTR double exponentiation algorithm and determine what information to gather to reconstruct the two input exponents. Our analysis and experimental results show that it takes U^{1. 25} tries, where U = max(a,b) on average to find the correct exponent pair (a,b). Using this result, we conclude that an adversary is expected to make U^{0. 625} tries on average until he/she finds the correct secret key used in XTR single exponentiation algorithm, which is based on the simultaneous XTR double exponentiation algorithm.

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This paper presents a brief history about the numbers. After some important definitions to understand the texts. Following, we encounter the world of prime numbers. This part is presented some important properties, findings and open problems. The study of these figures have managed to find some formulas to generate them, which are presented throughout the text. It presents some numbers especias such as Fermat primes, Mersene, Shopie German and others. Finally, we have an application that uses many properties presented.
Neste trabalho é apresentado um breve histórico sobre os números. Após, algumas definições importantes para compreensão dos textos. Seguindo, nos deparamos com o universo dos números primos. Nesta parte é apresentado algumas propriedades importantes, descobertas e problemas em aberto. O estudo sobre estes números já conseguiu encontrar algumas fórmulas para gerá-los, que são apresentadas no decorrer do texto. Apresenta-se alguns números especias, como os primos de Fermat, Mersene, Shopie German e outros. Por fim, temos uma aplicação que utiliza muitas propriedades apresentadas.

Tese de mestrado em Matemática para Professores, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2015
Os números e as suas regularidades desde sempre fascinaram os matemáticos. Ao longo dos tempos, a busca de provas ou refutações de várias conjeturas impulsionaram o avanço do conhecimento matemático, levando ao aparecimento da Teoria dos Números. Muitos foram os matemáticos de renome que, em diferentes momentos históricos, deram o seu contributo para esta evolução. Mesmo as antigas civilizações Babilónia e Egípcia tinham já conhecimentos sobre os números, as suas propriedades e regularidades, apesar das escassas referências escritas existentes não permitirem aferir rigorosamente o quão profundo era esse conhecimento. Já o mesmo não acontece com a civilização grega, cuja curiosidade, engenho e genialidade de alguns dos seus matemáticos se encontra bem documentada. O texto matemático mais importante da época grega foi, indubitavelmente, a obra de Euclides os “Elementos”, na qual, nos seus capítulos VII, VIII e IX, existem referências e provas de alguns resultados que revelam um profundo conhecimento da Teoria do Números, em particular, dos números perfeitos e dos números primos, cujas propriedades e regularidades apaixonaram os matemáticos em diferentes momentos. Com este trabalho, pretendemos realizar uma súmula dos resultados e conjeturas mais relevantes referentes ao processo que alicerçou o estudo dos números perfeitos desde a antiguidade até aos dias de hoje. Atualmente, a procura de números perfeitos resume-se a encontrar os denominados primos de Mersenne, isto é, primos da forma 2n−1, cujo trabalho do monge minimita Marin Mersenne mostrou estarem na base da factorização de todos os números perfeitos conhecidos. Tentaremos ainda fazer referência a alguns dos desafios, que atualmente persistem, referentes aos números perfeitos e às suas propriedades, assim como de algumas conjeturas que, apesar de experimentalmente corroboradas com recurso aos meios computacionais atuais, ainda carecem de prova ou refutação.
Numbers and their regularities forever fascinated mathematicians. Throughout the ages, the search for evidence or refutations of several conjectures boosted the advancement of mathematical knowledge, leading to the appearance of number theory. Many were the renowned mathematicians who, in different historical moments, contributed to this development. Even the ancient Babylonian and Egyptian civilizations had extensive knowledge about the numbers, their properties and regularities, in spite of the scarce written references which do not allow us to accurately gauge how deep was this knowledge. The same is not true with the Greek civilization, win which curiosity, resourcefulness and genius of some of their mathematicians is well documented. The most important mathematical text of that time was, undoubtedly, the work of Euclid's "Elements", in which, in chapter IX, there are references and evidence of some results which reveal a deep knowledge of the theory of numbers, in particular, of perfect and prime numbers, whose properties and regularities fascinated mathematicians at different times. With this work, we intend to present a collection of results and conjectures there were more relevant for the process that allowed the study of perfect numbers from antiquity to the present day. Currently, the demand for perfect numbers resumes itself to find what is now known as Mersenne primes, in honor of the monk Marin Mersenne that, among others results, showed that primes numbers that can be written in the form 2n -1 are factors in the factorization of all known perfect numbers. We also intend to make reference to some of the challenges that currently persist in the study of perfect numbers and their properties, as well as some conjectures that, although experimentally corroborated with current computational means, still lack proof or refutation.

碩士
輔仁大學
數學系碩士班
106
Abstract In this master thesis, we will briefly introduce some well-known results on Mersenne and Fermat numbers in Chapter 1. In Section 1.3, we will also introduce the Lucas-Lehmer sequences and outline their useful corresponding properties with complete proofs. In Chapter 2, we will study the Lucas-Lehmer primality test for Mersenne numbers. And in Chapter 3, we will study the Pepin{'}s primality test for Fermat numbers. Via Lucas-Lehmer sequences, both tests share a common nature. We will outline these ideas with complete discussions here.

碩士
國立臺灣大學
電機工程學研究所
101
Recent research on high-speed cryptography has been striving for performance by twiddling with instructions, but without an automated tool, writing fast software takes much precious labor effort. We present a tool with a simple interface for crypto developers to generate fast modular multiplication routines in a few keystrokes: you provide the prime as the modulus and it produces several candidate results or enumerates them all for benchmark. Specifically, we automatized the choice of number representation and the code generation for multiplication modulo a pseudo-Mesenne prime on ARM11, using the proposed convolved multiplication method, which interleaves multiplication and modular reduction. The high-quality code generated runs up to 16.4% faster than the convolved multiplication compiled by defacto-standard compilers such as gcc, and is 4 to 8 times faster than the GMP modular multiplication.

We investigate the origins of mathematical societies and journals. We argue that the origins of today’s professional societies and journals have their roots in the informal gatherings of mathematicians in 17th century Italy, France, and England. The small gatherings in these nations began as academies and after gaining government recognition and support, they became the ancestors of the professional societies that exist today. We provide a brief background on the influences of the Renaissance and Reformation before discussing the formation of mathematical academies in each country.

It has been said that Pythagoras discovered the perfect musical intervals by chance when he heard sounds of hammers striking an anvil at a nearby smithy. The sounds corresponded to the same intervals Pythagoras had been studying. He experimented with various instruments and apparatus to confirm what he heard. Math, and in particular, numbers are connected to music, he concluded. The discovery of musical intervals and the icon of the musical blacksmith have been familiar tropes in history, referenced in literary, musical, and visual arts. Countless authors since Antiquity have written about the story of the discovery, most often found in theoretical texts about music. However, modern scholarship has judged the narrative as a myth and a fabrication. Its refutation of the story is peculiar because modern scholarship has failed to disprove the nature of Pythagoras’s discovery with valid physical explanations. This report examines the structural elements of the story and traces its evolution since Antiquity to the early modern period to explain how an author interprets the narrative and why modern scholarship has deemed it a legend. The case studies of Nicomachus of Gerasa, Claudius Ptolemy, Boethius, and Marin Mersenne reveal not only how the story about Pythagoras’s discovery functions for each author, but also how the alterations in each version uncover an author’s views on music.
text