Prime Numbers

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54 Chapter 1 PRIMES! where p runs over primes. Conclude that there are infinitely many primes. One possible route is to establish the following intermediate steps: (1) Show that ⌊x⌋ n=1 1 n > ln x. (2) Show that 1 1 n = p≤x (1 − numbers n not divisible by any prime exceeding x. p )−1 , where the sum is over the natural 1.21. Use the multinomial (generalization of the binomial) theorem to show that for any positive integer u and any real number x>0, ⎛ 1 ⎝ u! ⎞u 1 ⎠ ≤ p 1 n , p≤x n≤x u where p runs over primes and n runs over natural numbers. Using this inequality with u = ⌊ln ln x⌋, show that for x ≥ 3, p≤x 1 ≤ ln ln x + O(lnlnlnx). p 1.22. By considering the highest power of a given prime that divides a given factorial, prove that N! = ∞ p k=1 ⌊N/pk⌋ , p≤N where the product runs over primes p. Then use the inequality N N N! > e (which follows from e N = ∞ k=0 N k /k! >N N /N !), to prove that p≤N ln p > ln N − 1. p − 1 Conclude that there are infinitely many primes. 1.23. Use the Stirling asymptotic formula N N √2πN N! ∼ e and the method of Exercise 1.22 to show that ln p =lnN + O(1). p p≤N Deduce that the prime-counting function π(x) satisfies π(x) =O(x/ ln x) and that if π(x) ∼ cx/ ln x for some number c, thenc =1.
1.5 Exercises 55 1.24. Derive from the Chebyshev Theorem 1.1.3 the following bounds on the n-th prime number pn for n ≥ 2: where C, D are absolute constants. Cnln n2 x for all x ≥ 31. p≤x
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Prime Numbers
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Richard Crandall Center for Advance
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Preface In this volume we have ende
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Preface ix Examples of computationa
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Contents Preface vii 1 PRIMES! 1 1.
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CONTENTS xiii 4.2.2 An improved n +
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CONTENTS xv 8.5.2 The Shor quantum
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2 Chapter 1 PRIMES! where exponents
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104 Chapter 2 NUMBER-THEORETICAL TO
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Chapter 3 RECOGNIZING PRIMES AND CO
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3.1 Trial division 119 d =3; while(
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3.2 Sieving 121 3.2 Sieving Sieving
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3.2 Sieving 123 this number’s ent
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3.2 Sieving 125 noticed that it was
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3.2 Sieving 127 } S = S \ (pS ∩ [
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3.3 Recognizing smooth numbers 129
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3.4 Pseudoprimes 131 } g =gcd(s, x)
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3.4 Pseudoprimes 133 Theorem 3.4.4.
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3.5 Probable primes and witnesses 1
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3.6 Lucas pseudoprimes 143 The Fibo
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3.6 Lucas pseudoprimes 145 Because
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3.6 Lucas pseudoprimes 147 use (3.1
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3.6 Lucas pseudoprimes 149 gcd(n, 2
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3.6 Lucas pseudoprimes 151 Theorem
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3.7 Counting primes 153 Label the c
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3.7 Counting primes 155 for b ≥ 2
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3.7 Counting primes 157 The heart o
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3.7 Counting primes 159 t =Im(s) ra
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3.7 Counting primes 161 Indeed, the
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3.8 Exercises 163 3.3. Prove that i
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3.8 Exercises 165 3.12. Show that a
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3.8 Exercises 167 3.28. Show that t
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3.9 Research problems 169 with W (n
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3.9 Research problems 171 3.50. The
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174 Chapter 4 PRIMALITY PROVING Rem
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176 Chapter 4 PRIMALITY PROVING sma
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178 Chapter 4 PRIMALITY PROVING Sin
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180 Chapter 4 PRIMALITY PROVING Let
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182 Chapter 4 PRIMALITY PROVING Rec
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184 Chapter 4 PRIMALITY PROVING (mo
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186 Chapter 4 PRIMALITY PROVING pol
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188 Chapter 4 PRIMALITY PROVING if
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190 Chapter 4 PRIMALITY PROVING 4.3
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192 Chapter 4 PRIMALITY PROVING j =
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194 Chapter 4 PRIMALITY PROVING The
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198 Chapter 4 PRIMALITY PROVING Rem
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200 Chapter 4 PRIMALITY PROVING pos
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202 Chapter 4 PRIMALITY PROVING Alg
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204 Chapter 4 PRIMALITY PROVING fac
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206 Chapter 4 PRIMALITY PROVING 196
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210 Chapter 4 PRIMALITY PROVING Say
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212 Chapter 4 PRIMALITY PROVING But
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214 Chapter 4 PRIMALITY PROVING for
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216 Chapter 4 PRIMALITY PROVING so
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218 Chapter 4 PRIMALITY PROVING (2)
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220 Chapter 4 PRIMALITY PROVING hav
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222 Chapter 4 PRIMALITY PROVING sho
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Chapter 5 EXPONENTIAL FACTORING ALG
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5.1 Squares 227 5.1.2 Lehman method
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5.2 Monte Carlo methods 229 That is
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5.2 Monte Carlo methods 231 It is c
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5.2 Monte Carlo methods 233 computi
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5.3 Baby-steps, giant-steps 235 cal
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5.4 Pollard p − 1 method 237 can
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5.6 Binary quadratic forms 239 f(jB
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5.6 Binary quadratic forms 241 so o
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5.6 Binary quadratic forms 243 equi
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5.6 Binary quadratic forms 245 is a
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5.6 Binary quadratic forms 247 In t
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5.6 Binary quadratic forms 249 of D
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5.7 Exercises 251 is completely rig
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5.7 Exercises 253 of each of these
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5.8 Research problems 255 5.17. Sho
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5.8 Research problems 257 modulo th
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5.8 Research problems 259 In judgin
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262 Chapter 6 SUBEXPONENTIAL FACTOR
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Chapter 7 ELLIPTIC CURVE ARITHMETIC
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7.1 Elliptic curve fundamentals 321
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7.2 Elliptic arithmetic 323 the poi
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7.2 Elliptic arithmetic 325 with EC
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7.2 Elliptic arithmetic 327 Algorit
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7.2 Elliptic arithmetic 329 Before
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7.2 Elliptic arithmetic 331 the “
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7.3 The theorems of Hasse, Deuring,
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7.4 Elliptic curve method 335 a ran
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7.4 Elliptic curve method 337 B1 =
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7.4 Elliptic curve method 339 facto
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7.4 Elliptic curve method 341 propa
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7.4 Elliptic curve method 343 As fo
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7.4 Elliptic curve method 345 if(1
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7.5 Counting points on elliptic cur
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7.6 Elliptic curve primality provin
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7.7 Exercises 375 7.4. As in Exerci
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7.7 Exercises 377 (some Bj equals A
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7.7 Exercises 379 This reduction ig
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7.8 Research problems 381 multiply-
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7.8 Research problems 383 highly ef
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7.8 Research problems 385 is prime.
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Chapter 8 THE UBIQUITY OF PRIME NUM
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8.1 Cryptography 389 is, if an orac
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8.1 Cryptography 391 Algorithm 8.1.
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8.1 Cryptography 393 just to genera
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8.1 Cryptography 395 where in the l
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8.2 Random-number generation 397 ar
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8.2 Random-number generation 399 Al
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8.2 Random-number generation 401 }
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8.2 Random-number generation 403 is
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8.3 Quasi-Monte Carlo (qMC) methods
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8.4 Diophantine analysis 415 [Tezuk
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8.4 Diophantine analysis 417 9262 3
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8.5 Quantum computation 419 We spea
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8.5 Quantum computation 421 three H
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8.5 Quantum computation 423 for a n
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8.6 Curious, anecdotal, and interdi
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8.7 Exercises 431 universal Golden
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8.7 Exercises 433 standards insist
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8.7 Exercises 435 of positive compo
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8.8 Research problems 437 element o
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8.8 Research problems 439 the Leveq
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8.8 Research problems 441 for every
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Chapter 9 FAST ALGORITHMS FOR LARGE
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9.1 Tour of “grammar-school” me
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9.2 Enhancements to modular arithme
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9.3 Exponentiation 457 Algorithm 9.
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9.3 Exponentiation 459 But there is
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9.3 Exponentiation 461 the benefit
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9.4 Enhancements for gcd and invers
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9.5 Large-integer multiplication 47
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9.6 Polynomial arithmetic 509 can i
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9.6 Polynomial arithmetic 511 Incid
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9.6 Polynomial arithmetic 513 where
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9.6 Polynomial arithmetic 515 such
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9.6 Polynomial arithmetic 517 Note
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9.7 Exercises 519 (3) Write out com
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9.7 Exercises 521 where “do” si
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9.7 Exercises 523 9.23. How general
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9.7 Exercises 525 two (and thus, me
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9.7 Exercises 527 0 2 +3 2 +0 2 is
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9.7 Exercises 529 9.49. In the FFT
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9.7 Exercises 531 adjustment step.
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9.7 Exercises 533 9.69. Implement A
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9.8 Research problems 535 less than
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9.8 Research problems 537 1.66), na
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9.8 Research problems 539 9.82. A c
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542 Appendix BOOK PSEUDOCODE Becaus
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544 Appendix BOOK PSEUDOCODE } ...;
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546 Appendix BOOK PSEUDOCODE Functi
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548 REFERENCES [Apostol 1986] T. Ap
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550 REFERENCES [Bernstein 2004b] D.
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552 REFERENCES [Buchmann et al. 199
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554 REFERENCES [Crandall 1997b] R.
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556 REFERENCES [Dudon 1987] J. Dudo
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558 REFERENCES [Goldwasser and Kili
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560 REFERENCES [Joe 1999] S. Joe. A
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562 REFERENCES [Lenstra 1981] H. Le
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564 REFERENCES [Montgomery 1987] P.
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566 REFERENCES [Oesterlé 1985] J.
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568 REFERENCES [Pomerance et al. 19
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570 REFERENCES [Schönhage and Stra
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572 REFERENCES [Sun and Sun 1992] Z
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574 REFERENCES [Weisstein 2005] E.
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Index ABC conjecture, 417, 434 abel
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INDEX 579 Catalan problem, ix, 415,
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INDEX 581 discrete arithmetic-geome
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INDEX 583 complex-field, 477 Cooley
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INDEX 585 Hajratwala, N., 23 Halber
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INDEX 587 Lehmer, D., 149, 152, 155
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INDEX 589 432, 447-450, 453, 458, 5
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INDEX 591 Preneel, B. (with De Win
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INDEX 593 Salomaa, A. (with Paun et
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INDEX 595 te Riele, H. (with van de
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INDEX 597 Yagle, A., 259, 499, 539
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