Prime Numbers

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114 Chapter 2 NUMBER-THEORETICAL TOOLS Establish the corollary that the number of pairs of consecutive quadratic residuesis(p − 5)/4, (p − 3)/4, respectively, as p ≡ 1, 3 (mod 4). Using the formula for N(a, b), prove that for every prime p the congruence x 2 + y 2 ≡−1(modp) is solvable. One satisfying aspect of the N(a, b) formula is the statistical notion that sure enough, if the Legendre symbol is thought of as generated by a “random coin flip,” there ought to be about p/4 occurrences of a given pair (±1, ±1). All of this makes sense: The Legendre symbol is in some sense random. But in another sense, it is not quite so random. Let us estimate a sum: sA,B = x , p A≤x
2.5 Research problems 115 half −1’s). We cannot even build up more than an O √ p ln p excess of one symbol over the other. But in a truly random coin-flip game, any pattern of 1’s and −1’s is allowed; and even if you constrain such a game to have equal numbers of 1’s and −1’s as does the Legendre-symbol game, there are still vast numbers of possible coin-flip sequences that cannot be symbol sequences. In some sense, however, the Pólya–Vinogradov inequality puts the Legendre symbol sequence smack in the middle of the distribution of possible sequences: It is what we might expect for a random sequence of coin flips. Incidentally, in view of the coin-flip analogy, what would be the expected value of the least quadratic nonresidue (mod p)? In this regard see Exercise 2.39. For a different kind of constraint on presumably random quadratic residues, see the remarks at the end of Exercise 2.29. 2.42. Here is a fascinating line of research: Using the age-old and glorious theory of the arithmetic–geometric mean (AGM), investigate the notion of what we might call a “discrete arithmetic–geometric mean (DAGM).” It was a tour de force of analysis, due to Gauss, Legendre, Jacobi, to conceive of the analytic AGM, which is the asymptotic fixed point of the elegant iteration a + b (a, b) ↦→ 2 , √ ab , that is, one replaces the pair (a, b) of real numbers with the new pair of arithmetic and geometric means, respectively. The classical AGM, then, is the real number c to which the two numbers converge; sure enough, (c, c) ↦→ (c, c) so the process tends to stabilize for appropriate initial choices of a and b. This scheme is connected with the theory of elliptic integrals, the calculation of π to (literally) billions of decimal places, and so on [Borwein and Borwein 1987]. But consider doing this procedure not on real numbers but on residues modulo a prime p ≡ 3 (mod 4), in which case an x (mod p) that has a square root always has a so-called principal root (and so an unambiguous choice of square root can be taken; see Exercise 2.25). Work out a theory of the DAGM modulo p. Perhaps you would want to cast √ ab as a principal root if said root exists, but something like a different principal root, say √ gab, for some fixed nonresidue g when ab is a nonresidue. Interesting theoretical issues are these: Does the DAGM have an interesting cycle structure? Is there any relation between your DAGM and the classical, analytic AGM? If there were any fortuitous connection between the discrete and analytic means, one might have a new way to evaluate with high efficiency certain finite hypergeometric series, as appear in Exercise 7.26.
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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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Page 130 and 131: Chapter 3 RECOGNIZING PRIMES AND CO
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
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Page 144 and 145: 3.4 Pseudoprimes 131 } g =gcd(s, x)
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Page 156 and 157: 3.6 Lucas pseudoprimes 143 The Fibo
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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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