Detail View

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.- A. Perott’s Proof.- B. Auric’s Proof.- C. Métrod’s Proof.- 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.- A. Fermat’s Little Theorem and Primitive Roots Modulo a Prime.- B. The Theorem of Wilson.- C. The Properties of Giuga, Wolstenholme, and Mann and Shanks.- D. The Power of a Prime Dividing a Factorial.- E. The Chinese Remainder Theorem.- F. Euler’s Function.- G. Sequences of Binomials.- H. Quadratic Residues.- 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.- A. Pseudoprimes in Base 2 (psp).- B. Pseudoprimes in Base a (psp(a)).- C. Euler Pseudoprimes in Base a (epsp(a)).- D. Strong Pseudoprimes in Base a (spsp(a)).- E. Somer Pseudoprimes.- IX. Carmichael Numbers.- X. Lucas Pseudoprimes.- A. Fibonacci Pseudoprimes.- B. Lucas Pseudoprimes (lpsp(P, Q)).- C. Euler-Lucas Pseudoprimes (elpsp(P, Q)) and Strong Lucas Pseudoprimes (slpsp(P, Q)).- D. Somer-Lucas Pseudoprimes.- E. Carmichael-Lucas Numbers.- XL Primality Testing and Large Primes.- A. The Cost of Testing.- B. More Primality Tests.- C. Primality Certification.- D. Fast Generation of Large Primes.- E. Titanic Primes.- F. Curious Primes.- XII. Factorization and Public Key Cryptography.- A. Factorization of Large Composite Integers.- B. 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.- A. Surveying the Problems.- B. Polynomials with Many Initial Prime Absolute Values.- C. The Prime-Producing Polynomials Races.- D. Primes of the Form m2 + 1.- 4 How Are the Prime Numbers Distributed?.- I. The Growth of—(x).- A. History Unfolding.- B. Sums Involving the Möbius Function.- C. Tables of Primes.- D. The Exact Value of—(x) and Comparison with x/(log x), Li(x), and R(x).- E. The Nontrivial Zeros of—(s).- F. Zero-Free Regions for—(s) and the Error Term in the Prime Number Theorem.- G. The Growth of—(s).- H. Some Properties of—(x).- II. The n th Prime and Gaps.- A. The n th Prime.- B. Gaps Between Primes.- Interlude.- III. Twin Primes.- Addendum on k-Tuples of Primes.- IV. Primes in Arithmetic Progression.- A. There Are Infinitely Many!.- B. The Smallest Prime in an Arithmetic Progression.- C. Strings of Primes in Arithmetic Progression.- V. Primes in Special Sequences.- VI. Goldbach’s Famous Conjecture.- VII. The Waring-Goldbach Problem.- A. Waring’s Problem.- B. The Waring-Goldbach Problem.- VIII. The Distribution of Pseudoprimes, Carmichael Numbers, and Values of Euler’s Function.- A. Distribution of Pseudoprimes.- B. Distribution of Carmichael Numbers.- C. Distribution of Lucas Pseudoprimes.- D. Distribution of Elliptic Pseudoprimes.- E. Distribution of 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 Values of Linear Polynomials.- II. Prime Values of Polynomials of Arbitrary Degree.- III. Polynomials with Many Successive Composite Values.- IV. Partitio Numerorum.- V. Some Probabilistic Estimates.- A. Distribution of Mersenne Primes.- B. The log log Philosophy.- VI. The Density of the Set of Regular Primes.- Conclusion.- The Pages That Couldn’t Wait.- Primes up to 10,000.- Index of Tables.- Index of Names.