The New Book of Prime Number Records

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×2n±1.- 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 10,000.- Index of Tables.- Index of Names.