The Theory Of Prime Number Classification

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 www.zwideprimes.com

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium series established to honour Professors A. J. Coleman and H. W. Ellis and to acknowledge their long-lasting interest in the quality of teaching undergraduate students. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book oj Records, reminded me very gently that the most "innumerate" people of the world are of a certain tribe in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes Morris, I'm from Brazil, but my book will contain numbers different from 'one.' " He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name), and consists of about 16 million digits of the number 11. "I assure you Morris, that in spite of the beauty of the apparent randomness of the decimal digits of 11, I'll be sure that my text will also include some words." Acknowledgment. The manuscript of this book was prepared on the word processor by Linda Nuttall. I wish to express my appreciation for the great care, speed, and competence of her work. Paulo Ribenboim CONTENTS Preface vii Guiding the Reader xiii Index of Notations xv Introduction Chapter 1. How Many Prime Numbers Are There? 3 I. Euclid's Proof 3 II.

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.

The Pulitzer Prize–winning magazine’s stories of mathematical explorations show that inspiration strikes haphazardly, revealing surprising solutions and exciting discoveries—with a foreword by James Gleick These stories from Quanta Magazine map the routes of mathematical exploration, showing readers how cutting-edge research is done, while illuminating the productive tension between conjecture and proof, theory and intuition. The stories show that, as James Gleick puts it in the foreword, “inspiration strikes willy-nilly.” One researcher thinks of quantum chaotic systems at a bus stop; another suddenly realizes a path to proving a theorem of number theory while in a friend's backyard; a statistician has a “bathroom sink epiphany” and discovers the key to solving the Gaussian correlation inequality. Readers of The Prime Number Conspiracy, says Quanta editor-in-chief Thomas Lin, are headed on “breathtaking intellectual journeys to the bleeding edge of discovery strapped to the narrative rocket of humanity's never-ending pursuit of knowledge.” Winner of the 2022 Pulitzer Prize for Explanatory Reporting, Quanta is the only popular publication that offers in-depth coverage of the latest breakthroughs in understanding our mathematical universe. It communicates mathematics by taking it seriously, wrestling with difficult concepts and clearly explaining them in a way that speaks to our innate curiosity about our world and ourselves. Readers of this volume will learn that prime numbers have decided preferences about the final digits of the primes that immediately follow them (the “conspiracy” of the title); consider whether math is the universal language of nature (allowing for “a unified theory of randomness”); discover surprising solutions (including a pentagon tiling proof that solves a century-old math problem); ponder the limits of computation; measure infinity; and explore the eternal question “Is mathematics good for you?” Contributors Ariel Bleicher, Robbert Dijkgraaf, Kevin Hartnett, Erica Klarreich, Thomas Lin, John Pavlus, Siobhan Roberts, Natalie Wolchover Copublished with Quanta Magazine

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.

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 Theory of Integer Classification, Distribution and Factorization The approach applied here is a unified classification system that uses an algebraic framework to establish a relationship and distribution system for integers. All numbers are defined algebraically through the universal number distribution theorem (UNDT). The advantage of the system is that: 1. It explains the gap theory of prime numbers, and the source of prime number randomness. 2. It revises and expands the concept and definition of a prime number. 3. It derives several primality test algorithms, and sieve structures. The theory has more than five ways of accurately testing for primality. 4. It derives the useful Integer Product Law that governs the distribution of the product between integers. It forms the basis for the development of the factorization engines. 5. It derives a general prime number factorization algorithm the structured factorization method (SFM framework). That is, factorization is precise and bounded in a given space. 6. It derives an algorithm that does both factorization and primality testing simultaneously (failure test approach). 7. It creates a factorization engine that runs on a transient database that is unique for a given number. This is the "6" factorization engine. This reduces time in looking for prime factors, and makes it not necessary to refer to a database of prime numbers when factorizing. 8. A powerful theorem (The Factor-2 Theorem) is established that allows for the development of the compressed factorization space using a concept of self-testing primality. 9. For the compressed factorization space, another factorization engine is defined leading to further compression of the factorization space. This is the "12" engine that is extremely efficient in creating the transient database for prime factors (turbo charged factorization). From classification, without a computer program but manually through an excel worksheet, the system was used to find factors of this number that was chosen at random. 100 785 423 193 771 = 3²x103x108 722 139 373 This was done in about ten minutes. The power of the classification system lies in the fact that there is no need for a database prime numbers since through the classification every number has a unique coordinate reference. Classification also yields the natural primality testing method, where through classification without any further computation, this 69 digit number is definitely not a prime number. You can show this in less than a minute! 789900541289753188975009123215467980768572971397525750310229845601223 The theory also defines the Merge operator that conceptualizes predictive factorization. Knowing the prime factors of one number, you can correctly predict two or more factors of a much larger random number without any further calculation. Through the same operator, you can also predict that one number will definitely have more prime factors than the other. Interesting and unusual! The book suggests new approaches in the understanding of integers in terms of classification, distribution and factorization. It is referred to as a unifying theory because it covers all the three areas. The book also provides a unique and clear understanding of the definition, behavior and distribution of prime numbers through the classification system. This leads to a revision of the Euclidean premise of infinity in regard to prime number distribution and the count function.

This 1952 book attempts to prove the Vinogradov-Goldbach theorem: that every sufficiently large odd number is the sum of three primes.

This book provides an introduction and overview of number theory based on the distribution and properties of primes. This unique approach provides both a firm background in the standard material as well as an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. Analytic number theory and algebraic number theory both receive a solid introductory treatment. The book’s user-friendly style, historical context, and wide range of exercises make it ideal for self study and classroom use.

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