Results 1  10
of
10
Some integer factorization algorithms using elliptic curves
 Australian Computer Science Communications
, 1986
"... Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order ..."
Abstract

Cited by 54 (13 self)
 Add to MetaCart
(Show Context)
Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1
Factorization Of The Tenth Fermat Number
 MATH. COMP
, 1999
"... We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factor ..."
Abstract

Cited by 23 (10 self)
 Add to MetaCart
(Show Context)
We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .
Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
Abstract

Cited by 17 (8 self)
 Add to MetaCart
(Show Context)
. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents
ACTA ARITHMETICA LX.1 (1991) Three additive cubic equations by
"... (Göttingen), ..."
(Show Context)
THE ψFUNCTION AND THE COMPLEXITY OF DIXON’S FACTORING ALGORITHM.
"... We will discuss the socalled ψfunction from analytic number theory and how it enters in the complexity analysis of a factoring algorithm. The function is defined (say) for real numbers x, y with 2 ≤ y ≤ x as follows: ψ(x, y) is defined as the number of n ∈ N with n ≤ x such that all prime divisors ..."
Abstract
 Add to MetaCart
(Show Context)
We will discuss the socalled ψfunction from analytic number theory and how it enters in the complexity analysis of a factoring algorithm. The function is defined (say) for real numbers x, y with 2 ≤ y ≤ x as follows: ψ(x, y) is defined as the number of n ∈ N with n ≤ x such that all prime divisors of n are ≤ y. Numbers n with all prime divisors ≤ y are often called ‘ysmooth’, or just ‘smooth ’ if it is clear from the connection what y is. Thus, ψ(x, y) is the number of ysmooth natural numbers ≤ x. Studying the asymptotic behavior of the ψfunction turns out to be the central point when analyzing the complexity of (most) modern factorization methods. 1. Some classical results. We will first mention some classical results pertaining to the ψfunction. This section is for background information only, and we will not go into the proofs. The proofs all involve rather heavy analytic number theoretic computations. For convenience one fixes the following notation: u:= log x log y, that is, we have y = x1/u.