## Prime numbers: a computational perspective. Second Edition (2005)

Citations: | 8 - 1 self |

### BibTeX

@MISC{Crandall05primenumbers:,

author = {Richard Crandall and Carl Pomerance and Richard Crandall and Carl Pomerance},

title = {Prime numbers: a computational perspective. Second Edition},

year = {2005}

}

### OpenURL

### Abstract

Cover illustration: The cover shows a magnified view—through a watchmaker’s loupe—of a very small portion of an actual poster giving the 7.2 million decimal digits of the prime 2 24036583-1. Ifthe entire one-meter tall poster were thus magnified, it would be the size of a baseball diamond.

### Citations

189 |
Solving sparse linear equations over finite fields
- Wiedemann
- 1986
(Show Context)
Citation Context ...merical analysis. These two, the conjugate gradient method and the Lanczos method, have been adapted to matrices with entries in a finite field. A third option is the coordinate recurrence method in [=-=Wiedemann 1986-=-]. This method is based on the Berlekamp–Massey algorithm for discovering the smallest linear recurrence relation in a sequence of finite field elements. Each of these methods can be accomplished with... |

148 | Cryptanalysis of short RSA secret exponents
- Wiener
- 1990
(Show Context)
Citation Context ...e can then readily compute an s such that s EB mod NB = x, so that Alice would possess a signed version of the harmful x. (3) Here we consider a small-private-exponent attack based on an analysis in [=-=Wiener 1990-=-]. Consider an RSA modulus N = pq with q<p<2q. Assume the usual condition ED mod ϕ(N) = 1, but we shall restrict the private exponent by D<N 1/4 /3. Show first that |N − ϕ(N)| < 3 √ N. Show then the e... |

146 |
Über die Gleichverteilung von Zahlen mod
- Weyl
- 1916
(Show Context)
Citation Context ... well below the trivial bound b − a. Thus, E(f; a, b, t) measures in a certain sense the distribution of fractional parts for the sequence (tf(n)), a<n≤ b. In fact, H. Weyl’s celebrated theorem (see [=-=Weyl 1916-=-]) asserts that the sequence (f(n)), n =1, 2,... is equidistributed modulo 1 if and only if for every integer h �= 0wehaveE(f;0,N,h)=o(N). Though distribution of fractional parts is a constant undercu... |

49 |
Average case complexity of multivariate integration
- Woźniakowski
- 1991
(Show Context)
Citation Context ... of discrepancy. Incidentally, one of the many fascinating theoretical results beyond Theorem 8.3.2 is the assessment of Wozniakowski of “average” case error bounds on the unit cube. As discussed in [=-=Wozniakowski 1991-=-], the statistical ensemble average—in an appropriately rigorous sense— of the integration error is closely related to discrepancy, verifying once and fors8.3 Quasi-Monte Carlo (qMC) methods 407 all t... |

24 |
The Great Internet Mersenne Prime Search. http://www.gimps.org
- Woltman, Kurowski
(Show Context)
Citation Context ...recent “largest known primes” performed thus: The primality proof/disproof for a single candidate 2 q − 1 required in 2004 about one CPUweek, on a typical modern PC (see continually updating website [=-=Woltman 2000-=-]). By contrast, a number of order 2 20000000 would have required, just a decade earlier, perhaps a decade of a typical PC’s CPU time! Of course, both machine and algorithm advances are responsible fo... |

21 | private communication - Zimmermann |

15 |
Zum letzten Fermat’schen Theorem
- Wieferich
- 1909
(Show Context)
Citation Context ...are primes such that 2 p−1 ≡ 1(modp 2 ), (1.14) such primes being called Wieferich primes. These special primes figure strongly in the so-called first case of Fermat’s “last theorem,” as follows. In [=-=Wieferich 1909-=-] it is proved that if x p + y p = z p , where p is a prime that does not divide xyz, thenp satisfies relation (1.14). Equivalently, we say that p is a Wieferich prime if the Fermat quotient qp(2) = 2... |

10 |
The evolutionary origins of periodical cicadas during ice ages. Am Nat 149
- Yoshimura
- 1997
(Show Context)
Citation Context ... on prime numbers. A pleasingly recreational source for interdisciplinary prime-number investigations is [Caldwell 1999]. In biology, prime numbers appear in contexts such as the following one, from [=-=Yoshimura 1997-=-]. We quote the author directly in order to show how prime numbers can figure into a field or a culture, without much of the standard number-theoretical language, rather with certain intuitive inferen... |

7 | Finding strong pseudoprimes to several bases - Zhang |

6 |
The InfoSleuth project
- Woelk, Tomlinson
- 1994
(Show Context)
Citation Context ...ariant SNFS has reached 248 decimal digits. The elliptic curve method (ECM) has now reached 59 decimal digits (for a prime factor that is not the largest in the number). Such records can be found in [=-=Zimmermann 2000-=-], a website that is continually updated. We provide a more extensive list of records below. Another interesting achievement has been the discovery of factors of various Fermat numbers Fn =22n+ 1 disc... |

5 |
Édouard Lucas and Primality Testing, volume 22
- Williams
- 1998
(Show Context)
Citation Context ...s from Euler’s criterion (2.6). ✷ Actually, Pepin gave his test with the number 5 in place of the number 3 (and with k ≥ 2). It was noticed by Proth and Lucas that one can use 3. In this regard, see [=-=Williams 1998-=-] and Exercise 4.5. As of this writing, the largest Fk for which the Pepin test has been used is F24. As discussed in Section 1.3.2, this number is composite, and in fact, so is every other Fermat num... |

5 |
congruential random-number generators with multiplier
- Wu, “Multiplicative
- 1997
(Show Context)
Citation Context ...d a certain success in passing many (but not all) experimental tests [Park and Miller 1988], [Press et al. 1996]. An interesting optimization of certain congruential generators has been forwarded in [=-=Wu 1997-=-]. The recursion is xn+1 = � (2 30 − 2 19 � )xn mod M61, where the fact of M61 being a Mersenne prime allows some rapid arithmetic. Algorithm 8.2.3 (Fast, 61-bit random generator). This algorithm prov... |

4 | Fast exponentiation using data compression
- Yacobi
- 1998
(Show Context)
Citation Context ...chains and Lucas ladders, as in Algorithm 3.6.7, interesting references being such as [Montgomery 1992b], [Müller 1998]. (6) Modern methods based on actual compression of exponent bit-streams, as in [=-=Yacobi 1999-=-]. The current section starts with basic binary ladders (and even for these, various options exist); then we turn to the windowing, alternative-base, and fixed-x ladders. 9.3.1 Basic binary ladders We... |

4 |
Fast Algorithms for Matrix Multiplication Using Pseudo-Number-Theoretic Transforms
- Yagle
- 1995
(Show Context)
Citation Context ...ee multiplies instead of four, and large-matrix multiplication can be endowed with its own special speedups, such as Strassen recursion [Crandall 1994b] and number-theoretical transform acceleration [=-=Yagle 1995-=-]; see Exercise 9.84. 5.29. Investigate the possibility of modifying the polynomial evaluation method of Pollard and Strassen for application to the factorization of Fermat numbers Fn =22n + 1. Since ... |

4 |
The dierences between consecutive primes
- Yu
- 1996
(Show Context)
Citation Context ... O � t89/560+ɛ� . The Lindelöf hypothesis is the conjecture that ζ(1/2 +it) =O(tɛ ) for any ɛ>0. This conjecture also has consequences for the distribution of primes, such as the following result in [=-=Yu 1996-=-]: If pn denotes the n-th prime, then on the Lindelöf hypothesis, � pn≤x (pn+1 − pn) 2 = x 1+o(1) . The best that is known unconditionally is that the sum is O � x23/18+ɛ� for any ɛ>0, a result of D. ... |

3 | Factoring integers before computers - Williams, Shallit |

3 |
On Waring’s problem in finite fields
- Winterhof
- 1998
(Show Context)
Citation Context ...um methods appear in many references, e.g., [Hardy 1966], [Nathanson 1996], [Vaughan 1997].) Incidentally, many Waring-type questions for finite fields have been completely resolved; see for example [=-=Winterhof 1998-=-]. 9.81. Is there a way to handle large convolutions without DFT, by using the kind of matrix idea that underlies Algorithm 9.5.7? That is, you would be calculating a convolution in small pieces, with... |

3 |
1/f noise in the distribution of prime numbers
- Wolf
- 1997
(Show Context)
Citation Context ...erence one can find further references to p-adic quantum gravity and p-adic Einstein-relativistic equations. Physicists have from time to time even performed “prime number experiments.” For example, [=-=Wolf 1997-=-] takes a signal, call it x = (x0,x1,...,xN−1), where a component xj is the count of primes over some interval. Specifically, xj = π((j +1)M) − π(jM), where M is some fixed interval length. Then is co... |

2 | A One-Parameter Quadratic-Base Version of the Baillie-PSW - Zhang - 2002 |

1 |
Private communication
- Wellin
- 1998
(Show Context)
Citation Context ...arandache–Wellin prime is w128 = 23571113171923 ...719 . How many decimal digits does w128 have? Incidentally, large as this example is, yet larger such primes (at least, probable primes!) are known [=-=Wellin 1998-=-], [Weisstein 2005]. 1.87. Show the easy result that if k primes each larger than k lie in arithmetic progression, then the common difference d is divisible by every prime not exceeding k. Find a long... |

1 | Factorization of the numbers of the form m 3 - REFERENCES - 1998 |