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The Development of Prime Number Theory

Author: Wladyslaw Narkiewicz

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 457

ISBN-13: 3662131579

1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's Elemen ta, where we find a proof of their infinitude, now regarded as canonical. One feels that Euclid's argument has its place in The Book, often quoted by the late Paul ErdOs, where the ultimate forms of mathematical arguments are preserved. Proofs of most other results on prime number distribution seem to be still far away from their optimal form and the aim of this book is to present the development of methods with which such problems were attacked in the course of time. This is not a historical book since we refrain from giving biographical details of the people who have played a role in this development and we do not discuss the questions concerning why each particular person became in terested in primes, because, usually, exact answers to them are impossible to obtain. Our idea is to present the development of the theory of the distribu tion of prime numbers in the period starting in antiquity and concluding at the end of the first decade of the 20th century. We shall also present some later developments, mostly in short comments, although the reader will find certain exceptions to that rule. The period of the last 80 years was full of new ideas (we mention only the applications of trigonometrical sums or the advent of various sieve methods) and certainly demands a separate book.

The Prime Number Theorem

Author: G. J. O. Jameson

Publisher: Cambridge University Press

Published: 2003-04-17

Total Pages: 266

ISBN-13: 9780521891103

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At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce 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. The prime number theorem tells us what this formula is and it is indisputably one of the great classical theorems of mathematics. This textbook gives an introduction to the prime number theorem suitable for advanced undergraduates and beginning graduate students. The author's aim is to show the reader how the tools of analysis can be used in number theory to attack a 'real' problem, and it is based on his own experiences of teaching this material.

Additive Theory of Prime Numbers

Author: Luogeng Hua

Publisher: American Mathematical Soc.

Published: 2009-12-04

Total Pages: 206

ISBN-13: 0821849425

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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 Obsession

Author: John Derbyshire

Publisher: Joseph Henry Press

Published: 2003-04-15

Total Pages: 446

ISBN-13: 0309141257

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In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark â€" a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic â€" defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark â€" the Riemann Hypothesis â€" that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows â€" subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many â€" the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof â€" and those who have been consumed by it.

Number Theory and Its History

Author: Oystein Ore

Publisher: Courier Corporation

Published: 2012-07-06

Total Pages: 400

ISBN-13: 0486136434

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Unusually clear, accessible introduction covers counting, properties of numbers, prime numbers, Aliquot parts, Diophantine problems, congruences, much more. Bibliography.

The Distribution of Prime Numbers

Author: Albert Edward Ingham

Publisher: Cambridge University Press

Published: 1990-09-28

Total Pages: 140

ISBN-13: 9780521397896

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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.

Rational Number Theory in the 20th Century

Author: Władysław Narkiewicz

Publisher: Springer Science & Business Media

Published: 2011-09-02

Total Pages: 659

ISBN-13: 0857295322

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The last one hundred years have seen many important achievements in the classical part of number theory. After the proof of the Prime Number Theorem in 1896, a quick development of analytical tools led to the invention of various new methods, like Brun's sieve method and the circle method of Hardy, Littlewood and Ramanujan; developments in topics such as prime and additive number theory, and the solution of Fermat’s problem. Rational Number Theory in the 20th Century: From PNT to FLT offers a short survey of 20th century developments in classical number theory, documenting between the proof of the Prime Number Theorem and the proof of Fermat's Last Theorem. The focus lays upon the part of number theory that deals with properties of integers and rational numbers. Chapters are divided into five time periods, which are then further divided into subject areas. With the introduction of each new topic, developments are followed through to the present day. This book will appeal to graduate researchers and student in number theory, however the presentation of main results without technicalities will make this accessible to anyone with an interest in the area.

How Mathematics Happened

Author: Peter S. Rudman

Publisher: Prometheus Books

Published: 2009-12-30

Total Pages: 316

ISBN-13: 1615921761

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In this fascinating discussion of ancient mathematics, author Peter Rudman does not just chronicle the archeological record of what mathematics was done; he digs deeper into the more important question of why it was done in a particular way. Why did the Egyptians use a bizarre method of expressing fractions? Why did the Babylonians use an awkward number system based on multiples of 60? Rudman answers such intriguing questions, arguing that some mathematical thinking is universal and timeless. The similarity of the Babylonian and Mayan number systems, two cultures widely separated in time and space, illustrates the argument. He then traces the evolution of number systems from finger counting in hunter-gatherer cultures to pebble counting in herder-farmer cultures of the Nile and Tigris-Euphrates valleys, which defined the number systems that continued to be used even after the invention of writing. With separate chapters devoted to the remarkable Egyptian and Babylonian mathematics of the era from about 3500 to 2000 BCE, when all of the basic arithmetic operations and even quadratic algebra became doable, Rudman concludes his interpretation of the archeological record. Since some of the mathematics formerly credited to the Greeks is now known to be a prior Babylonian invention, Rudman adds a chapter that discusses the math used by Pythagoras, Eratosthenes, and Hippasus, which has Babylonian roots, illustrating the watershed difference in abstraction and rigor that the Greeks introduced. He also suggests that we might improve present-day teaching by taking note of how the Greeks taught math. Complete with sidebars offering recreational math brainteasers, this engrossing discussion of the evolution of mathematics will appeal to both scholars and lay readers with an interest in mathematics and its history.

A Primer of Analytic Number Theory

Author: Jeffrey Stopple

Publisher: Cambridge University Press

Published: 2003-06-23

Total Pages: 404

ISBN-13: 9780521012539

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An undergraduate-level 2003 introduction whose only prerequisite is a standard calculus course.

Number Theory

Author:

Publisher: Academic Press

Published: 1986-05-05

Total Pages: 449

ISBN-13: 0080873324

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This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used. We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them. Most of the questions that are examined in this book are connected with the theory of diophantine equations - that is, with the theory of the solutions in integers of equations in several variables. However, we also consider questions of other types; for example, we derive the theorem of Dirichlet on prime numbers in arithmetic progressions and investigate the growth of the number of solutions of congruences.