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The Life of Primes in 37 Episodes

About this Title

Jean-Marie De Koninck, Université Laval, Québec City, QC, Canada and Nicolas Doyon, Université Laval, Québec City, QC, Canada

Publication: AMS Non-Series Monographs
Publication Year: 2021; Volume 139
ISBNs: 978-1-4704-6489-9 (print); 978-1-4704-6537-7 (online)
DOI: https://doi.org/10.1090/mbk/139

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Table of Contents

Front/Back Matter

• View this volume's front and back matter

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 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

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