Prime Numbers

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3.7 Counting primes 153 Label the consecutive primes p1,p2,p3,...,wherep1 =2,p2 =3,p3 =5, etc. Let φ(x, y) =#{1 ≤ n ≤ x : each prime dividing n is greater than y}. Thus φ(x, pa) is the number of integers left unmarked in the sieve of Eratosthenes, applied to the interval [1,x], after sieving with p1,p2,...,pa. Since sieving up to √ x leaves only the number 1 and the primes in ( √ x, x], we have π(x) − π √ x +1=φ x, √ x . One could easily use this idea to compute π(x), the time taking O(x ln ln x) operations and, if the sieve is segmented, taking O x 1/2 ln x space. (We shall begin suppressing ln x and ln ln x factors for simplicity, sweeping them under a rather large rug of O(x ɛ ). It will be clear that each x ɛ could be replaced, with a little more work, with a small power of logarithm and/or double logarithm.) A key thought is that the sieve not only allows us to count the primes, it also identifies them. If it is only the count we are after, then perhaps we can be speedier. We shall partition the numbers counted by φ(x, y) by the number of prime factors they have, counted with multiplicity. Let φk(x, y) =#{n ≤ x : n has exactly k prime factors, each exceeding y}. Thus, if x ≥ 1, φ0(x, y) is1,φ1(x, y) is the number of primes in (y, x], φ2(x, y) is the number of numbers pq ≤ x where p, q areprimeswithy
154 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES allowed to run over the interval [p, x/p], and so the number of choices for q is π(x/p) − π(p) + 1. Thus, φ2(x, x 1/3 )= (π(x/p) − π(p)+1) The last sum is = π(x 1/3 )
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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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4 Chapter 1 PRIMES! of the most rec
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6 Chapter 1 PRIMES! decision bit) o
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8 Chapter 1 PRIMES! 1.1.4 Asymptoti
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10 Chapter 1 PRIMES! naive subrouti
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12 Chapter 1 PRIMES! x π(x) 10 2 1
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14 Chapter 1 PRIMES! Erdős-Turán
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16 Chapter 1 PRIMES! Let’s hear i
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18 Chapter 1 PRIMES! Conjecture 1.2
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20 Chapter 1 PRIMES! It is not hard
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22 Chapter 1 PRIMES! 1.3 Primes of
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24 Chapter 1 PRIMES! 9.5.18 and Alg
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26 Chapter 1 PRIMES! Mersenne conje
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28 Chapter 1 PRIMES! Again, as with
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30 Chapter 1 PRIMES! then taken aga
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32 Chapter 1 PRIMES! McIntosh has p
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34 Chapter 1 PRIMES! have identitie
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36 Chapter 1 PRIMES! This theorem i
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38 Chapter 1 PRIMES! conjectured. T
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40 Chapter 1 PRIMES! Definition 1.4
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42 Chapter 1 PRIMES! on the left co
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44 Chapter 1 PRIMES! sums. These su
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46 Chapter 1 PRIMES! Vinogradov’s
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48 Chapter 1 PRIMES! been beyond re
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50 Chapter 1 PRIMES! prime? What is
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52 Chapter 1 PRIMES! This kind of c
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54 Chapter 1 PRIMES! where p runs o
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56 Chapter 1 PRIMES! While the prim
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58 Chapter 1 PRIMES! Exercise 1.35.
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60 Chapter 1 PRIMES! this recreatio
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62 Chapter 1 PRIMES! so that A3(x)
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64 Chapter 1 PRIMES! Conclude that
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66 Chapter 1 PRIMES! implies that
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68 Chapter 1 PRIMES! the Riemann-Si
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70 Chapter 1 PRIMES! such sums can
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72 Chapter 1 PRIMES! Cast this sing
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74 Chapter 1 PRIMES! 10 10 . The me
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76 Chapter 1 PRIMES! These numbers
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78 Chapter 1 PRIMES! Next, as for q
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80 Chapter 1 PRIMES! (see [Bach 199
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82 Chapter 1 PRIMES! If one invokes
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84 Chapter 2 NUMBER-THEORETICAL TOO
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100 Chapter 2 NUMBER-THEORETICAL TO
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Page 132 and 133: 3.1 Trial division 119 d =3; while(
Page 134 and 135: 3.2 Sieving 121 3.2 Sieving Sieving
Page 136 and 137: 3.2 Sieving 123 this number’s ent
Page 138 and 139: 3.2 Sieving 125 noticed that it was
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Page 144 and 145: 3.4 Pseudoprimes 131 } g =gcd(s, x)
Page 146 and 147: 3.4 Pseudoprimes 133 Theorem 3.4.4.
Page 148 and 149: 3.5 Probable primes and witnesses 1
Page 156 and 157: 3.6 Lucas pseudoprimes 143 The Fibo
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Page 162 and 163: 3.6 Lucas pseudoprimes 149 gcd(n, 2
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Page 170 and 171: 3.7 Counting primes 157 The heart o
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204 Chapter 4 PRIMALITY PROVING fac
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206 Chapter 4 PRIMALITY PROVING 196
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208 Chapter 4 PRIMALITY PROVING The
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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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