Corpus ID: 21341476

Factorizations of Cunningham numbers with bases 13 to 99

  title={Factorizations of Cunningham numbers with bases 13 to 99},
  author={R. Brent and P. L. Montgomery and H. Riele},
This Report updates the tables of factorizations of a^n +- 1 for 13 < a < 100, previously published as CWI Report NM-R9212 (June 1992) and updated in CWI Report NM-R9419 (Update 1, September 1994) and CWI Report NM-R9609 (Update 2, March 1996). A total of 951 new entries in the tables are given here. The factorizations are now complete for n < 76, and there are no composite cofactors smaller than 10^102. 

Tables and Topics from this paper


FACTORIZATIONS OF a n ± 1, 13 a < 100
As an extension of the “Cunningham” tables [4, 5] we present tables of factorizations of a n ± 1 for 13 a < 100. The exponents n satisfy a n < 10 255 if a < 30, and n 100 if a 30. The factorizationsExpand
Speeding the Pollard and elliptic curve methods of factorization
Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1Expand
Factoring Integers with Large-Prime Variations of the Quadratic Sieve
Experiments show that for the Cray C90 implementations PPMPQS beats PMPZS for numbers of more than 80 digits, and that this crossover point goes down with the amount of available central memory. Expand
The multiple polynomial quadratic sieve
A modification, due to Peter Montgomery, of Pomerance's Quadratic Sieve for factoring large integers is discussed along with its implementation. Using it, allows factorization with over an order ofExpand
A survey of modern integer factorization algorithms
Introduction An integer n is said to be a prime number or simply prime if the only divisors of n are and n There are in nitely many prime numbers the rst four being and If n and n is not prime then nExpand
On computing factors of cyclotomic polynomials
  • R. Brent
  • Mathematics, Computer Science
  • ArXiv
  • 2010
It is shown how the coefficients of the cyclotomic polynomial An(x), . . . , Dn(x) are polynomials with integer coefficients which can be computed by simple algorithms which require O(n) arithmetic operations and work over the integers. Expand
Factorization of the tenth Fermat number
  • R. Brent
  • Computer Science, Mathematics
  • Math. Comput.
  • 1999
The complete factorization of the tenth Fermat number F10 by the elliptic curve method (ECM) is described, which is a product of four prime factors with 8, 10, 40 and 252 decimal digits. Expand
An FFT extension of the elliptic curve method of factorization
Factorization of arbitrary integers is believed to be a hard problem. The Elliptic Curve Method (ECM), discovered by Hendrik Lenstra, Jr. in 1985, is the best known method for finding 20- to 30-digitExpand
Centrum Voor Wiskunde En Informatica
An estimate is given of the size of a solution n 2 N of the inequality (an+ b) < (an), gcd(a; b) = 1. Experiments indicate that this gives a useful indication of the size of the minimal solution.Expand
Factoring integers with elliptic curves
This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2. Expand