1 How Many Prime Numbers Are There?. - I. Euclid's Proof. - II. Goldbach Did It Too . - III. Euler's Proof. - IV. Thue's Proof. - V. Three Forgotten Proofs. - VI. Washington's Proof. - VII. Fürstenberg's Proof. - VIII. Euclidean Sequences. - IX. Generation of Infinite Sequences of Pairwise Relatively Prime Integers. - 2 How to Recognize Whether a Natural Number Is a Prime. - I. The Sieve of Eratosthenes. - II. Some Fundamental Theorems on Congruences. - III. Classical Primality Tests Based on Congruences. - IV. Lucas Sequences. - V. Primality Tests Based on Lucas Sequences. - VI. Fermat Numbers. - VII. Mersenne Numbers. - VIII. Pseudoprimes. - IX. Carmichael Numbers. - X. Lucas Pseudoprimes. - XL Primality Testing and Large Primes. - XII. Factorization and Public Key Cryptography. - 3 Are There Functions Defining Prime Numbers?. - I. Functions Satisfying Condition (a). - II. Functions Satisfying Condition (b). - III. Functions Satisfying Condition (c). - IV. Prime-Producing Polynomials. - 4 How Are the Prime Numbers Distributed?. - I. The Growth of ?(x). - II. The n th Prime and Gaps. - Interlude. - III. Twin Primes. - Addendum on k-Tuples of Primes. - IV. Primes in Arithmetic Progression. - V. Primes in Special Sequences. - VI. Goldbach's Famous Conjecture. - VII. The Waring-Goldbach Problem. - VIII. The Distribution of Pseudoprimes Carmichael Numbers and Values of Euler's Function. - 5 Which Special Kinds of Primes Have Been Considered?. - I. Regular Primes. - II. Sophie Germain Primes. - III. Wieferich Primes. - IV. Wilson Primes. - V. Repunits and Similar Numbers. - VI. Primes with Given Initial and Final Digits. - VII. Numbers k×2n1. - VIII. Primes and Second-Order Linear Recurrence Sequences. - IX. The NSW Primes. - 6 Heuristic and Probabilistic Results about Prime Numbers. - I. Prime Valuesof Linear Polynomials. - II. Prime Values of Polynomials of Arbitrary Degree. - III. Polynomials with Many Successive Composite Values. - IV. Partitio Numerorum. - V. Some Probabilistic Estimates. - Conclusion. - The Pages That Couldn't Wait. - Primes up to 10000. - Index of Tables. - Index of Names. Language: English