Prime Numbers – Old mysteries, new records

Name: Prime Numbers – Old mysteries, new records

Author(s): e Paulo Ribenboim

Pages: 317

Publication: IMPA, 2020

ISBN: 978-65-89124-03-0

Edition: 3

Preface to the second edition

Preface to the first edition

A short guide for the reader

Thanks

Translator’s note

1 How many prime numbers are there?
1.1 Euclid’s demonstration
1.2 Goldbach also demonstrated
1.3 Euler’s demonstration
1.4 Thue’s demonstration
1.5 Three forgotten demonstrations
1.5.1 Perott’s demonstration
1.5.2 Auric’s demonstration
1.5.3 Métrod’s demonstration
1.6 Washington’s demonstration
1.7 Furstenberg’s demonstration

2 How to solve prime numbers?
2.1 Eratosthenes’ sieve
2.2 Some fundamental theorems about congruence
2.2.1 Fermat’s little theorem and the primitive roots modulo a prime number
2.2.2 Wilson’s theorem
2.2.3 Giuga and Wolstenholme properties
2.2.4 The power of a prime number dividing a factorial
2.2.5 The Chinese theorem
2.2.6 The Euler function
2.2.7 Successes of binomials
2.2.8 Quadratic residuals
2.3 Classic primality tests
2.4 Lucas’ successions
2.5 Primality tests based on Lucas successions
2.6 Fermat’s numbers
2.7 Mersenne numbers
2.8 Pseudoprime numbers
2.8.1 Pseudoprime numbers in base 2 (psp)
2.8.2 Pseudoprime numbers in base a (psp( a))
2.8.3 Pseudoprime Euler numbers in base a (epsp( a))
2.8.4 Strong pseudoprime numbers in base a (spsp( a))
2.9 The Carmichel numbers
2.10 Lucas pseudoprime numbers
2.10.1 Fibonacci pseudoprimes
2.10.2 Lucas pseudoprime numbers (lpsp( P, Q))
2.10.3 Euler-Lucas pseudoprimes (elpsp( P, Q)) and strong Lucas pseudoprimes (slpsp( P, Q))
2.10.4 Carmichel-Lucas numbers
2.11 Primality and factoring
2.11.1 The cost of testing
2.11.2 Other primality tests
2.11.3 The titanic and curious cousins
2.11.4 Factoring
2.11.5 Public key cryptography

3 Are there functions that define prime numbers?
3.1 Functions satisfying condition (A)
3.2 Functions satisfying condition (B)
3.3 Functions satisfying condition (C)

4 How are prime numbers distributed?
4.1 The growth of π( x)
4.1.1 History
4.1.2 Sums using the Möbius function
4.1.3 The distribution of Euler function values
4.1.4 Prime number tables
4.1.5 Estimation and exact value of π( x) and comparison with x/log x, Li( x) and R( x)
4.1.6 The non-trivial zeros of ζ( s)
4.1.7 Non-zero regions of ζ( s) and the error term of the prime number theorem
4.2 The nth prime number and the spacing between successive primes
4.2.1 Some properties of π( x)
4.2.2 The nth prime number
4.2.3 Spacing between consecutive prime numbers
4.3 Twin prime numbers
4.4 k-tuples of prime numbers
4.5 Cousins in arithmetic progression
4.5.1 There are infinitely many of them!
4.5.2 The smallest prime number in an arithmetic progression
4.5.3 Successions of prime numbers in arithmetic progression
4.6 Goldbach’s famous conjecture
4.7 Pseudoprimes and Carmichel numbers
4.7.1 Distribution of pseudoprime numbers
4.7.2 Distribution of Carmichel numbers
4.7.3 Distribution of Lucas pseudoprime numbers

5 Which particular prime numbers were studied?
5.1 Regular primes
5.2 Sophie Germain’s cousins
5.3 The Wieferich cousins
5.4 Wilson’s cousins
5.5 Repunits and similar numbers
5.6 Prime numbers of the form k x ^bn ± 1
5.6.1 Numbers of the form k x 2 ⁿ + 1 and Sierpinski numbers
5.6.2 Numbers of the form k x 2 ⁿ – 1 and Riesel numbers
5.6.3 Numbers of the form k x ^bn ± 1 with k ≥ 1 and b > 2
5.6.4 Generalized Fermat numbers
5.6.5 The Cullen numbers
5.6.6 Woodall’s numbers
5.7 Prime numbers in second-order linear recursive sequences

6 Heuristics and probabilistic results on prime numbers
6.1 Prime numbers values of linear polynomials
6.2 Prime numbers values of polynomials of arbitrary degree
6.3 Polynomials having many successive composite values
6.4 Partitio Numerorum

Appendix

Conclusion

Bibliography
General references
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6

Prime numbers up to 10000
Index of notations
List of tables
Index of records
Index of authors
Index of references