Download or read book On solutions of some of unsolved problems in number theory specifically on the distribution of primes written by Ahmad Sabihi and published by Infinite Study. This book was released on 2023-01-01 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt: We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre’s, Andrica’s, Oppermann’s, Brocard’s, Cramer’s, Shanks’, and five Smarandache’s conjectures. We make use of both Firoozbakht’s conjecture (which recently proved by the author) and Kourbatov’s theorem on the distribution of and gaps between consecutive primes. These latter conjecture and theorem play an essential role in our methods for proving these famous conjectures. In order to prove Shanks’ conjecture, we make use of Panaitopol’s asymptotic formula for π(x) as well.
Download or read book Unsolved Problems in Number Theory written by R.K. Guy and published by Springer. This book was released on 1981-11-30 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: Second edition sold 2241 copies in N.A. and 1600 ROW. New edition contains 50 percent new material.
Download or read book The Prime Number Theorem written by G. J. O. Jameson and published by Cambridge University Press. This book was released on 2003-04-17 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Irregularities in the Distribution of Prime Numbers written by János Pintz and published by Springer. This book was released on 2018-07-04 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book The Prime Numbers and Their Distribution written by Gerald Tenenbaum and published by American Mathematical Soc.. This book was released on 2000 with total page 137 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book The Fundamental Theorem of Arithmetic written by Demetrios P Kanoussis Ph D and published by Independently Published. This book was released on 2023-12-20 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: A prime number is a natural number greater than 1, which cannot be expressed as a product of two smaller natural numbers. Stated differently, a prime number is a number divided only by 1 and by itself. No other natural number can divide a prime number. For example, the numbers: 2, 3, 5, 7, 11, 13, 17, 19, etc, are prime numbers. All the other numbers are called composite numbers. For example, the numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc, are composite numbers. The prime number 7 is divided by 1 and 7 only, while the composite number 12 is divided by 1, by 12, but also, by 2, and by 3, and by 4, and by 6. Prime numbers, despite their simple definition, still hide many secrets, and their intensive study has been the subject of many eminent mathematicians, for more than 25 centuries. Prime numbers and their properties were studied, for the first time, by the ancient Greeks. Euclid defined a prime number as "a number measured by a unit alone" and also gave the first proof that there are infinitely many prime numbers, (around 300 BC). Eratosthenes of Cyrene, around 200 BC, devised a procedure, (an algorithm), to find all the prime numbers less than a given natural number, (the sieve of Eratosthenes). The main problem with prime numbers is that the way they appear is highly irregular. There is no known formula, or method, to predict the next prime number from the preceding one. Great mathematicians, Gauss, Euler and Riemann among others, have made important contributions to the prime numbers, but, the main problem of finding a pattern as how the prime numbers are distributed, still, remains unsolved. There are also, many other unsolved problems concerning the prime numbers. One of them is the Goldbach's conjecture, that every even number greater than 2 can be expressed as the sum of two primes. Another famous unsolved problem is the twin prime conjecture, that there are infinitely many pairs of primes that differ by 2, (like 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc). Other unsolved problems concerning prime numbers and some historical remarks, like, Euclid's theorem, the sieve of Eratosthenes of Cyrene, Fermat numbers, etc, are discussed in section 3-4. Primes are of fundamental importance in number theory, because of the fundamental theorem of arithmetic: every natural number greater than 1, can be expressed as a product of prime numbers, and this product representation is unique. In this little book we provide a rigorous proof of this theorem and present some of its applications, (like for instance, to find the number of divisors and the sum of the divisors of any natural number, the perfect numbers and the amicable numbers). The material covered in the book is shown analytically in the table of contents. A comprehensive introduction to the prime numbers and some of its more important properties are presented, which are essential for the proof of the fundamental theorem. Solving problems with prime numbers is not an easy task. Most of the times, the solution requires ingenuity and imagination. Some other times, when a direct method of approach does not lead to the solution of the problem, we may proceed by contradiction, which is a popular and powerful method when working with prime numbers. The careful reader will find that many of the solved exercises have been solved by contradiction. The book contains 60 illustrative, solved exercises and 115 problems for solution.
Download or read book Exploring Numbers Beyond Primes written by N.B. Singh and published by N.B.Singh. This book was released on with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Exploring Numbers Beyond Primes" is a comprehensive and accessible introduction to the fascinating world of number theory, designed specifically for absolute beginners. This book takes readers on a captivating journey through the mysteries of prime numbers and Diophantine equations, offering clear explanations and engaging examples along the way. From the historical origins of number theory to modern approaches and future directions, each chapter provides a step-by-step exploration of key concepts, supported by vivid descriptions and relatable analogies. Whether you're a curious novice or an aspiring mathematician, "Exploring Numbers Beyond Primes" invites you to discover the beauty and wonder of numbers, inspiring a lifelong passion for mathematical inquiry.
Download or read book The Distribution of Prime Numbers written by Albert Edward Ingham and published by Cambridge University Press. This book was released on 1990-09-28 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Prime Objective written by Andy Brown and published by . This book was released on 2020-07-19 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: Prime numbers and their properties have been studied by many mathematicians over hundreds of years, including such names as Euclid, Euler, Gauss and Riemann. It was Bernard Riemann who made a significant contribution in the form of a short paper submitted in 1859 entitled "On the number of primes less than a given magnitude". His experience in the field of Complex numbers was brought to bear on the behaviour and distribution of prime numbers and was responsible for the Riemann Hypothesis an unsolved problem in number theory, having defied a proof for more than 150 years. This book attempts to follow all the derivations in detail starting from established formulae and tries to avoid phrases such as "it can be easily shown that" which leave people like me wondering just how it can be easily shown that.... Riemann sprinkled a few of these phrases in his paper, but fortunately there are many papers, books and information available from on-line sources related to this topic, particularly around the famous Riemann Hypothesis, a famous conjecture made by Riemann in the course of this work.The exploration of the topic starts with a look at Prime Numbers, those building blocks of the set of whole numbers. A particularly relevant function which was explored by Euler is the Zeta function. Euler derived a formula that linked the set of primes with the set of natural numbers using the Zeta function. Riemann found that this function is also closely related to the distribution of Prime Numbers.The derived functional equation is evaluated to show how well it maps onto the distribution of the Primes. I have used home grown c-programs to reproduce results and check how well the formula follows the prime distribution.
Download or read book An Infinity Of Unsolved Problems Concerning A Function In The Number Theory written by FLORENTIN SMARANDACHE and published by Infinite Study. This book was released on with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt: W.Sierpinski has asserted to an international conference that if mankind lasted for ever and numbered the unsolved problems, then in the long run all these unsolved problems would be solved.
Download or read book The Development of Prime Number Theory written by Wladyslaw Narkiewicz and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Solved and Unsolved Problems in Number Theory written by Daniel Shanks and published by American Mathematical Society. This book was released on 2024-01-24 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment. This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.
Download or read book The Life of Primes in 37 Episodes written by Jean-Marie De Koninck and published by American Mathematical Soc.. This book was released on 2021-05-19 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about the life of primes. Indeed, once they are defined, primes take on a life of their own and the mysteries surrounding them begin multiplying, just like living cells reproduce themselves, and there seems to be no end to it. This monograph takes the reader on a journey through time, providing an accessible overview of the numerous prime number theory problems that mathematicians have been working on since Euclid. Topics are presented in chronological order as episodes. These include results on the distribution of primes, from the most elementary to the proof of the famous prime number theorem. The book also covers various primality tests and factorisation algorithms. It is then shown how our inability to factor large integers has allowed mathematicians to create today's most secure encryption method. Computer science buffs may be tempted to tackle some of the many open problems appearing in the episodes. Throughout the presentation, the human side of mathematics is displayed through short biographies that give a glimpse of the lives of the people who contributed to the life of primes. Each of the 37 episodes concludes with a series of problems (many with solutions) that will assist the reader in gaining a better understanding of the theory.
Download or read book Prime Numbers The Holy Grail Of Mathematics written by Thamer Naouech and published by Thamer Naouech. This book was released on 2020-10-20 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is undeniable how prime numbers are one of the most beautiful and fascinating topics in mathematics. But what are prime numbers? Are they only numbers that are divisible by 1 and themselves, or do they have another interesting hidden face?Throughout history, the mystery of prime numbers has challenged the greatest minds in mathematics starting from Euclid of Alexandria to Fermat, Euler, Gauss, and Erdős,… who attempted to solve the puzzling problem of primes. The achievements they realized and the secrets they revealed can only assert how deep the concept of prime numbers is. Starting from how prime numbers exist in nature, and how they are of great use in modern cryptography on which our daily life completely depends, the author travels in the holy kingdom of primes diving into some conjectures involving those special numbers. From the Riemann Hypothesis and the well-known zeta function, he explains how a note in the margin turned to be Fermat’s Last Theorem, one of the most important problems in the history of mathematics. From Mersenne Primes, he gets to the twin primes, those shining little stars in the blue sky of primes. And from Euclid’s proof of the infinite number of primes he gets to a hidden pattern in the distribution of primes discovered by Stanisław Ulam and called the Ulam Spiral. After this little trip, you will know, dear reader, why prime numbers deserve to be called "the holy grail of mathematics".
Download or read book The Music of the Primes written by Marcus Du Sautoy and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recounts the history of prime numbers (numbers divisible only by one and themselves) which has implications beyond pure mathematics. Euclid, Ramanujan, Odlyzko, and the formation of RSA encryption are discussed, as well as the state of contemporary research on the topic.
Download or read book Additive Theory of Prime Numbers written by Luogeng Hua and published by American Mathematical Soc.. This book was released on 2009-12-04 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
