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Full Description
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.
Contents
Counting primes, the road to the prime number theorem: An infinite family
The search for large primes
The great insight of Legendre and Gauss
Euler, the visionary
Dirichlet's theorem
The Berstrand postulate and the Chebyshev theorem
Riemannn shows the way
Connecting the zeta function to the prime counting function
The intriguing Riemann hypothesis
Mertens' theorems
Couting the number of primes, from Meissel to today
Hadamard and de la Vallee Poussin stun the world
An elementary proof of the prime number theorem
Counting primes, beyond the prime number theorem: Sieve methods
Prime clusters
Primes in arithmetic progression
Small and large gaps between consecutive primes
Irregularities in the distribution of primes
Exceptional sets of primes
The birth of probabilistic number theory
The multiplicative structure of integers
Generalized prime number systems
Is it a prime?: Establishing if a given integer is prime or not
The Lucas and Pepin primality tests
Those annoying Carmichael numbers
The Lucas-Lehmer primality test for Mersenne numbers
The probabilistic Miller-Rabin primality test
The deterministic AKS primality test
Finding the prime factors of a given integer: The Fermat factorisation algorithm
From the Fermat factorisation algorithm to the quadratic sieve
The Pollard $p$-1 factorisation algorithm
The Pollard Rho factorisaction algorithm
Two factorisation methods based on modern algebra
Algebraic factorisation
Measuring and comparing the speed of various algorithms
Making good use of the primes and moving forward: Cryptography, from Julius Caesar to the RSA cryptosystem
The present and future life of primes
Appendix A. A time line of some key results on prime numbers
Appendix B. Hints, sketches and solutions to a selection of problems
Appendix C. Basic results from number theory, algebra and analysis
Bibliography
Notation and symbols
Index of short biographies
Index of subjects
