The New Book of Prime Number Records
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1 Paulo Ribenboim The New Book of Prime Number Records Springer
2 Contents Preface Guiding the Reader Index of Notations ix xv xvii Introduction 1 CHARTER 1 How Many Prime Numbers Are There? 3 I. Euclid's Proof 3 II. Goldbach Did It Too! 4 III. Euler's Proof 6 IV. Thue's Proof 7 V. Three Forgotten Proofs 8 A. Perott's Proof 8 B. Auric's Proof 9 C. Metrod's Proof 9 VI. Washingtons Proof 9 VII. Fürstenberg's Proof 10 VIII. Euclidean Sequences 11 IX. Generation of Infinite Sequences of Pairwise Relatively Prime Integers 17 CHAPTER 2 How to Recognize Whether a Natural Number Is a Prime 19 I. The Sieve of Eratosthenes 20 II. Some Fundamental Theorems on Congruences 21 A. Fermat's Little Theorem and Primitive Roots Modulo a Prime 22 B. The Theorem of Wilson 25 C. The Properties of Giuga, Wolstenholme, and Mann and Shanks 28 D. The Power of a Prime Dividing a Factorial 30 E. The Chinese Remainder Theorem 33 F. Euler's Function 34 G. Sequences of Binomials 42 H. Quadratic Residues 46 III. Classical Primality Tests Based on Congruences 48 IV. Lucas Sequences 53 V. Primality Tests Based on Lucas Sequences 74 VI. Fermat Numbers 83 xi
3 Xll Contents VII. Mersenne Numbers 90 VIII. Pseudoprimes 103 A. Pseudoprimes in Base 2 (psp) 105 B. Pseudoprimes in Base a (psp(a)) 108 C. Euler Pseudoprimes in Base a (epsp(a)) 112 D. Strong Pseudoprimes in Base a (spsp(fl)) 113 E. Somer Pseudoprimes 117 IX. Carmichael Numbers 118 X. Lucas Pseudoprimes 126 A. Fibonacci Pseudoprimes 127 B. Lucas Pseudoprimes (lpsp(p, ß)) 129 C. Euler-Lucas Pseudoprimes (elpsp(p, ß)) and Strong Lucas Pseudoprimes (slpsp(p, ß)) 130 D. Somer-Lucas Pseudoprimes 131 E. Carmichael-Lucas Numbers 132 XI. Primality Testing and Large Primes 135 A. The Costof Testing 136 B. More Primality Tests 139 C. Primality Certification 153 D. Fast Generation of Large Primes 156 E. Titanic Primes 157 F. Curious Primes 159 XII. Factorization and Public Key Cryptography 162 A. Factorization of Large Composite Integers 163 B. Public Key Cryptography 172 CHAPTER 3 Are There Functions Defining Prime Numbers? 179 I. Functions Satisfying Condition (a) 179 II. Functions Satisfying Condition (b) 186 III. Functions Satisfying Condition (c) 187 IV. Prime-Producing Polynomials 196 A. Surveying the Problems 196 B. Polynomials with Many Initial Prime Absolute Values 197 C. The Prime-Producing Polynomials Races 203 D. Primes of the Form m CHAPTER 4 How Are the Prime Numbers Distributed? 213 I. TheGrowthof;r(x) 214 A. History Unfolding 215 B. Sums Involving the Möbius Function 229 C. Tables of Primes 233 D. The Exact Value of 7r(x) and Comparison with x/(log x), Li(x), and R(x) 235 E. The Nontrivial ZerosofC(s) 239 F. Zero-Free Regions for (s) and the Error Term in the Prime Number Theorem 243
4 Contents xiii G. TheGrowthofC(s) 245 H. Some Properties of n{x) 247 II. The nth Prime and Gaps 248 A. The nth Prime 248 B. Gaps Between Primes 250 Interlude 258 III. Twin Primes 259 Addendum onfc-tuplesof Primes 265 IV. Primes in Arithmetic Progression 265 A. There Are Infinitely Many! 265 B. The Smallest Prime in an Arithmetic Progression 277 C. Strings of Primes in Arithmetic Progression 284 V. Primes in Special Sequences 288 VI. Goldbach's Famous Conjecture 291 VII. The Waring-Goldbach Problem 299 A. Waring's Problem 300 B. The Waring-Goldbach Problem 310 VIII. The Distribution of Pseudopnmes, Carmichael Numbers, and Values of Euler's Function 311 A. Distribution of Pseudoprimes 311 B. Distribution of Carmichael Numbers 314 C. Distribution of Lucas Pseudoprimes 317 D. Distribution of Elliptic Pseudoprimes 318 E. Distribution of Values of Euler's Function 319 CHAPTER 5 Which Special Kinds of Primes Have Been Considered? 323 I. Regulär Primes 323 II. Sophie Germain Primes 329 III. Wieferich Primes 333 IV. Wilson Primes 346 V. Repunits and Similar Numbers 350 VI. Primes with Given Initial and Final Digits 355 VII. Numbers k x 2" ± VIII. Primes and Second-Order Linear Recurrence Sequences 361 IX. The NSW Primes 367 CHAPTER 6 Heuristic and Probabilistic Results about Prime Numbers 371 I. Prime Values of Linear Polynomials 372 II. Prime Values of Polynomials of Arbitrary Degree 386 III. Polynomials with Many Successive Composite Values 400 IV. Partitio Numerorum 403 V. Some Probabilistic Estimates 411 A. Distribution of Mersenne Primes 411 B. The log log Philosophy 413 VI. The Density of the Set of Regulär Primes 414
5 xiv Contents Conclusion Bibliography The Pages That Couldn't Wait Primes up to 10,000 Index of Tables Index of Names Subject Index
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