The Prime Number Theorem


Author: G. J. O. Jameson

Publisher: Cambridge University Press

ISBN: 9780521891103

Category: Mathematics

Page: 252

View: 819

The prime numbers appear to be distributed in a very irregular way amongst the integers, but the prime number theorem provides a simple formula that tells us (in an approximate but well-defined sense) how many primes we can expect to find that are less than any integer we might choose. This is indisputably one of the the great classical theorems of mathematics. Suitable for advanced undergraduates and beginning graduates, this textbook demonstrates how the tools of analysis can be used in number theory to attack a famous problem.

Prime Numbers

A Computational Perspective


Author: Richard Crandall,Carl B. Pomerance

Publisher: Springer Science & Business Media

ISBN: 0387289798

Category: Mathematics

Page: 597

View: 9329

Bridges the gap between theoretical and computational aspects of prime numbers Exercise sections are a goldmine of interesting examples, pointers to the literature and potential research projects Authors are well-known and highly-regarded in the field

The Distribution of Prime Numbers


Author: A. E. Ingham,Albert Edward Ingham

Publisher: Cambridge University Press

ISBN: 9780521397896

Category: Mathematics

Page: 114

View: 7985

Originally published in 1934, this volume presents the theory of the distribution of the prime numbers in the series of natural numbers. Despite being long out of print, it remains unsurpassed as an introduction to the field.

Prime Numbers, Friends Who Give Problems

A Trialogue with Papa Paulo


Author: Paulo Ribenboim

Publisher: World Scientific

ISBN: 9814725838

Category: Mathematics

Page: 336

View: 2124

'" Prime Numbers, Friends Who Give Problems is written as a trialogue, with two persons who are interested in prime numbers asking the author, Papa Paulo, intelligent questions. Starting at a very elementary level, the book advances steadily, covering all important topics of the theory of prime numbers, up to the most famous problems. The humorous conversations and the inclusion of a back-story add to the uniqueness of the book. Concepts and results are also explained with great care, making the book accessible to a wide audience. Contents:What are Prime Numbers?Division is Harder than MultiplicationAnother Paulo! Is a Dialogue of Three Possible?How Natural Numbers are Made Out of PrimesTell Me Which is the Largest Prime?Trying Hard to Find PrimesA Formula, A Formula, PleasePaulo Came with a LassoBeautiful Old Elementary ArithmeticThe Old Man Still KnowsCan You Tell Me All About Congruences?Homework CheckedTesting for Primality and FactorizationFermat Numbers are Friendly. Are They Primes?This World is PerfectUnfriendly Numbers from a Friend of Fermat''sPaying My DebtMoney and PrimesSecret MessagesNew Numbers and FunctionsPrinceps GaussGathering ForcesThe After "Math" of GaussPrimes After Dinner: Bad Dreams?Primes in Arithmetic ProgressionSelling PrimesThe Great Prime MysteriesMysteries in Sequences: More But Not AllThe End and the Beginning Readership: College students, high school teachers and beginners interested in number theory and important facts about prime numbers. "'

Additive Theory of Prime Numbers


Author: Luogeng Hua

Publisher: American Mathematical Soc.

ISBN: 0821849425

Category: Mathematics

Page: 190

View: 3770

Loo-Keng Hua was a master mathematician, best known for his work using analytic methods in number theory. In particular, Hua is remembered for his contributions to Waring's Problem and his estimates of trigonometric sums. Additive Theory of Prime Numbers is an exposition of the classic methods as well as Hua's own techniques, many of which have now also become classic. An essential starting point is Vinogradov's mean-value theorem for trigonometric sums, which Hua usefully rephrases and improves. Hua states a generalized version of the Waring-Goldbach problem and gives asymptotic formulas for the number of solutions in Waring's Problem when the monomial $x^k$ is replaced by an arbitrary polynomial of degree $k$. The book is an excellent entry point for readers interested in additive number theory. It will also be of value to those interested in the development of the now classic methods of the subject.

Prime Numbers

The Most Mysterious Figures in Math


Author: David Wells

Publisher: John Wiley & Sons

ISBN: 0471718920

Category: Mathematics

Page: 288

View: 2287

A fascinating journey into the mind-bending world of prime numbers Cicadas of the genus Magicicada appear once every 7, 13, or 17 years. Is it just a coincidence that these are all prime numbers? How do twin primes differ from cousin primes, and what on earth (or in the mind of a mathematician) could be sexy about prime numbers? What did Albert Wilansky find so fascinating about his brother-in-law's phone number? Mathematicians have been asking questions about prime numbers for more than twenty-five centuries, and every answer seems to generate a new rash of questions. In Prime Numbers: The Most Mysterious Figures in Math, you'll meet the world's most gifted mathematicians, from Pythagoras and Euclid to Fermat, Gauss, and Erd?o?s, and you'll discover a host of unique insights and inventive conjectures that have both enlarged our understanding and deepened the mystique of prime numbers. This comprehensive, A-to-Z guide covers everything you ever wanted to know--and much more that you never suspected--about prime numbers, including: * The unproven Riemann hypothesis and the power of the zeta function * The "Primes is in P" algorithm * The sieve of Eratosthenes of Cyrene * Fermat and Fibonacci numbers * The Great Internet Mersenne Prime Search * And much, much more

Prime Numbers and Computer Methods for Factorization


Author: Hans Riesel

Publisher: Springer Science & Business Media

ISBN: 9780817682989

Category: Mathematics

Page: 464

View: 6348

From the original hard cover edition: In the modern age of almost universal computer usage, practically every individual in a technologically developed society has routine access to the most up-to-date cryptographic technology that exists, the so-called RSA public-key cryptosystem. A major component of this system is the factorization of large numbers into their primes. Thus an ancient number-theory concept now plays a crucial role in communication among millions of people who may have little or no knowledge of even elementary mathematics. Hans Riesel’s highly successful first edition of this book has now been enlarged and updated with the goal of satisfying the needs of researchers, students, practitioners of cryptography, and non-scientific readers with a mathematical inclination. It includes important advances in computational prime number theory and in factorization as well as re-computed and enlarged tables, accompanied by new tables reflecting current research by both the author and his coworkers and by independent researchers. The book treats four fundamental problems: the number of primes below a given limit, the approximate number of primes, the recognition of primes and the factorization of large numbers. The author provides explicit algorithms and computer programs, and has attempted to discuss as many of the classically important results as possible, as well as the most recent discoveries. The programs include are written in PASCAL to allow readers to translate the programs into the language of their own computers. The independent structure of each chapter of the book makes it highly readable for a wide variety of mathematicians, students of applied number theory, and others interested in both study and research in number theory and cryptography. ​

Prime numbers


Author: William John Ellison,Fern Ellison

Publisher: John Wiley & Sons


Category: Mathematics

Page: 416

View: 2980

Lukasiewicz's Logics and Prime Numbers


Author: A. S. Karpenko

Publisher: Luniver Press

ISBN: 0955117038

Category: Mathematics

Page: 168

View: 7012

Is there any link between the doctrine of logical fatalism and prime numbers? What do logic and prime numbers have in common? The book adopts truth-functional approach to examine functional properties of finite-valued Łukasiewicz logics Łn+1. Prime numbers are defined in algebraic-logical terms (Finn's theorem) and represented as rooted trees. The author designs an algorithm which for every prime number n constructs a rooted tree where nodes are natural numbers and n is a root. Finite-valued logics Kn+1 are specified that they have tautologies if and only if n is a prime number. It is discovered that Kn+1 have the same functional properties as Łn+1 whenever n is a prime number. Thus, Kn+1 are 'logics' of prime numbers. Amazingly, combination of logics of prime numbers led to uncovering a law of generation of classes of prime numbers. Along with characterization of prime numbers author also gives characterization, in terms of Łukasiewicz logical matrices, of powers of primes, odd numbers, and even numbers.

The Prime Numbers and Their Distribution


Author: Gerald Tenenbaum,Michel Mendès France

Publisher: American Mathematical Soc.

ISBN: 0821816470

Category: Mathematics

Page: 115

View: 5811

One notable new direction this century in the study of primes has been the influx of ideas from probability. The goal of this book is to provide insights into the prime numbers and to describe how a sequence so tautly determined can incorporate such a striking amount of randomness. The book opens with some classic topics of number theory. It ends with a discussion of some of the outstanding conjectures in number theory. In between are an excellent chapter on the stochastic properties of primes and a walk through an elementary proof of the Prime Number Theorem. This book is suitable for anyone who has had a little number theory and some advanced calculus involving estimates. Its engaging style and invigorating point of view will make refreshing reading for advanced undergraduates through research mathematicians.

Prime Numbers and Computer Methods for Factorization


Author: H. Riesel

Publisher: Springer Science & Business Media

ISBN: 1475710895

Category: Mathematics

Page: 464

View: 308

In this book the author treats four fundamental and apparently simple problems. They are: the number of primes below a given limit, the ap proximate number of primes, the recognition of prime numbers and the factorization of large numbers. A chapter on the details of the distribution of the primes is included as well as a short description of a recent applica tion of prime numbers, the so-called RSA public-key cryptosystem. The author is also giving explicit algorithms and computer programs. Whilst not claiming completeness, the author has tried to give all important results known, including the latest discoveries. The use of computers has in this area promoted a development which has enormously enlarged the wealth of results known and that has made many older works and tables obsolete. As is often the case in number theory, the problems posed are easy to understand but the solutions are theoretically advanced. Since this text is aimed at the mathematically inclined layman, as well as at the more advanced student, not all of the proofs of the results given in this book are shown. Bibliographical references in these cases serve those readers who wish to probe deeper. References to recent original works are also given for those who wish to pursue some topic further. Since number theory is seldom taught in basic mathematics courses, the author has appended six sections containing all the algebra and number theory required for the main body of the book.

Irregularities in the Distribution of Prime Numbers

From the Era of Helmut Maier's Matrix Method and Beyond


Author: János Pintz,Michael Th. Rassias

Publisher: Springer

ISBN: 3319927779

Category: Mathematics

Page: 217

View: 369

This volume presents research and expository papers highlighting the vibrant and fascinating study of irregularities in the distribution of primes. Written by an international group of experts, contributions present a self-contained yet unified exploration of a rapidly progressing area. Emphasis is given to the research inspired by Maier’s matrix method, which established a newfound understanding of the distribution of primes. Additionally, the book provides an historical overview of a large body of research in analytic number theory and approximation theory. The papers published within are intended as reference tools for graduate students and researchers in mathematics.

The Solitude of Prime Numbers


Author: Paolo Giordano

Publisher: Random House

ISBN: 1409082385

Category: Fiction

Page: 384

View: 6943

A prime number is inherently a solitary thing: it can only be divided by itself, or by one; it never truly fits with another. Alice and Mattia also move on their own axes, alone with their personal tragedies. As a child Alice's overbearing father drove her first to a terrible skiing accident, and then to anorexia. When she meets Mattia she recognises a kindred spirit, and Mattia reveals to Alice his terrible secret: that as a boy he abandoned his mentally-disabled twin sister in a park to go to a party, and when he returned, she was nowhere to be found. These two irreversible episodes mark Alice and Mattia's lives for ever, and as they grow into adulthood their destinies seem irrevocably intertwined. But then a chance sighting of a woman who could be Mattia's sister forces a lifetime of secret emotion to the surface. A meditation on loneliness and love, The Solitude of Prime Numbers asks, can we ever truly be whole when we're in love with another?

Closing the Gap

The Quest to Understand Prime Numbers


Author: Vicky Neale

Publisher: Oxford University Press

ISBN: 0191092436

Category: Mathematics

Page: 208

View: 3406

In 2013, a little known mathematician in his late 50s stunned the mathematical community with a breakthrough on an age-old problem about prime numbers. Since then, there has been further dramatic progress on the problem, thanks to the efforts of a large-scale online collaborative effort of a type that would have been unthinkable in mathematics a couple of decades ago, and the insight and creativity of a young mathematician at the start of his career. Prime numbers have intrigued, inspired and infuriated mathematicians for millennia. Every school student studies prime numbers and can appreciate their beauty, and yet mathematicians' difficulty with answering some seemingly simple questions about them reveals the depth and subtlety of prime numbers. Vicky Neale charts the recent progress towards proving the famous Twin Primes Conjecture, and the very different ways in which the breakthroughs have been made: a solo mathematician working in isolation and obscurity, and a large collaboration that is more public than any previous collaborative effort in mathematics and that reveals much about how mathematicians go about their work. Interleaved with this story are highlights from a significantly older tale, going back two thousand years and more, of mathematicians' efforts to comprehend the beauty and unlock the mysteries of the prime numbers.

Simplicity Instinct

Why Prime Numbers Are Elusive!


Author: Philip Gervase Jackson

Publisher: Strategic Book Publishing

ISBN: 1622125096

Category: Self-Help

Page: 197

View: 2240

Come take a journey of exploration and discovery into the underworld of complexity, searching for simple understanding of our world and everything in it! Author Philip Gervase Jackson’s search for answers started at the age of five and within two years he was debating things others routinely took for granted. As an adult, Mr. Jackson used his new, evolving observations to develop a wonderfully powerful problem-solving method that helped him make discoveries in many different fields including mathematics and software design. As he applied the critical thinking skills he had nurtured, he realized that this problem-solving method is indeed universal - it can be applied to anything. Simplicity Instinct: Why Prime Numbers are Elusive! is a how-to book that shares all the opportunities provided by this method. Life often presents hurdles and barriers, but the search for deeper understanding will win over, every time.

On the Theory of Numbers

Prime Numbers and Enlightenment


Author: M.W.R. Karskens

Publisher: BookBaby

ISBN: 9082794004

Category: Mathematics

Page: 60

View: 9923

On the Theory of Numbers explains how numbers fit together.

The Theory of Prime Number Classification


Author: Zwide Mbulawa

Publisher: Xlibris Corporation

ISBN: 9781453598948

Category: Mathematics

Page: 261

View: 3197

The Theory of Prime Number Classification This is an expository work of mathematical research into the prime numbers based on pattern methodology and classification techniques. As a comprehensive research into the classification systems for prime numbers, it address the following: „X Why prime numbers are regular yet random. „X What are the building blocks of prime numbers „X What is the framework for prime number generation This is done by developing the following classification systems: „X The Prime Root Classification. All prime numbers are constituted by roots, which are defined as the building blocks of the prime number. „X The Positional Classification. A two dimensional prime number space is defined that allows certain types of distribution analysis of primes to be made, deriving count functions and establishing the mean property of primes „X The Delta Classification of Primes. This classification creates prime families in terms of gaps. Prime gaps are found to have positive, negative and a steady gap acceleration. „X The Gap Theory Classification. All prime gaps and prime number behavior are based on Gap 2, Gap 4 and Gap 6. This then develops a classification system. Using the above classification systems, and defining a special function, a theory of prime number generation is then suggested, where this leads to the development of an algebraic sieve for finding prime numbers. The algebraic sieve contains all the relevant information about prime numbers, including how gaps widen, and prime number patterns. Consequently, it is then used to address the problem of finding a proof for the twin prime conjecture. As an expository work, the book also shares personal experiences and thoughts with regard to the research, and the development of expository mathematics. A program for prime number classification is available at