Softcover ISBN:  9781470464899 
Product Code:  MBK/139 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Electronic ISBN:  9781470465377 
Product Code:  MBK/139.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 

Book Details2021; 329 ppMSC: Primary 11;
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.
ReadershipUndergraduate and graduate students and researchers interested in prime numbers.

Table of 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 Vallée 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 Pépin primality tests

Those annoying Carmichael numbers

The LucasLehmer primality test for Mersenne numbers

The probabilistic MillerRabin 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


Additional Material

Reviews

This is a beautiful book, very well written and edited. It should appeal to number theorists as well as interested mathematicians in other fields. It is also a rich source of supplementary readings for any undergraduate or graduate course in number theory.
Karl Dilcher, Dalhousie University


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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.
Undergraduate and graduate students and researchers interested in prime numbers.

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 Vallée 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 Pépin primality tests

Those annoying Carmichael numbers

The LucasLehmer primality test for Mersenne numbers

The probabilistic MillerRabin 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

This is a beautiful book, very well written and edited. It should appeal to number theorists as well as interested mathematicians in other fields. It is also a rich source of supplementary readings for any undergraduate or graduate course in number theory.
Karl Dilcher, Dalhousie University