Results 1  10
of
12
Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials
 in Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, Fields Institute
, 2003
"... Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although there i ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although there is no irreducible trinomial of degree r.
Pseudorandom number generation by padic ergodic transformations: an addendum

, 2004
"... The paper study counterdependent pseudorandom number generators based on mvariate (m> 1) ergodic mappings of the space of 2adic integers Z2. The sequence of internal states of these generators is defined by the recurrence law xi+1 = H B i (xi) mod 2 n, whereas their output sequence is zi = F B i ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
The paper study counterdependent pseudorandom number generators based on mvariate (m> 1) ergodic mappings of the space of 2adic integers Z2. The sequence of internal states of these generators is defined by the recurrence law xi+1 = H B i (xi) mod 2 n, whereas their output sequence is zi = F B i (xi) mod 2 n; here xj, zj are mdimensional vectors over Z2. It is shown how the results obtained for a univariate case could be extended to a multivariate case.
20 years of ECM
 In Proceedings of the 7th Algorithmic Number Theory Symposium (ANTS VII
, 2006
"... Abstract. The Elliptic Curve Method for integer factorization (ECM) was invented by H. W. Lenstra, Jr., in 1985 [14]. In the past 20 years, many improvements of ECM were proposed on the mathematical, algorithmic, and implementation sides. This paper summarizes the current stateoftheart, as implem ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Abstract. The Elliptic Curve Method for integer factorization (ECM) was invented by H. W. Lenstra, Jr., in 1985 [14]. In the past 20 years, many improvements of ECM were proposed on the mathematical, algorithmic, and implementation sides. This paper summarizes the current stateoftheart, as implemented in the GMPECM software.
Integer Factorization
, 2006
"... Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the sc ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.” But what exactly is the problem? It turns out that there are many different factorization problems, as we will discuss in this paper.
Some longperiod random number generators using shifts and xors
 13th CTAC06 Proceedings
, 2006
"... and xors ..."
Two new factors of Fermat numbers
, 1997
"... Abstract. We report the discovery of new 27decimal digit factors of the thirteenth and sixteenth Fermat numbers. Each of the new factors was found by the elliptic curve method. After division by the new factors and other known factors, the quotients are seen to be composite numbers with 2391 and 19 ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
Abstract. We report the discovery of new 27decimal digit factors of the thirteenth and sixteenth Fermat numbers. Each of the new factors was found by the elliptic curve method. After division by the new factors and other known factors, the quotients are seen to be composite numbers with 2391 and 19694 decimal digits respectively. 1.
Three New Factors of Fermat Numbers
 Math. Comp
, 2000
"... We report the discovery of a new factor for each of the Fermat numbers F 13 ,F 15 ,F 16 . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We report the discovery of a new factor for each of the Fermat numbers F 13 ,F 15 ,F 16 . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694 decimal digits respectively. 1.
Almost Irreducible and Almost Primitive Trinomials
 in Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, Fields Institute
, 2003
"... Consider polynomials over GF(2). We de ne almost irreducible and almost primitive polynomials, explain why they are useful, and give some examples and conjectures relating to them. 2 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Consider polynomials over GF(2). We de ne almost irreducible and almost primitive polynomials, explain why they are useful, and give some examples and conjectures relating to them. 2
Efficient SIMD arithmetic modulo a Mersenne number
 In IEEE Symposium on Computer Arithmetic – ARITH20
, 2011
"... Abstract—This paper describes carryless arithmetic operations modulo an integer 2 M − 1 in the thousandbit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract—This paper describes carryless arithmetic operations modulo an integer 2 M − 1 in the thousandbit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game consoles a new record was set for the elliptic curve method for integer factorization.
On the Infinitude of Some Special Kinds of Primes — — Dedicated to the memory of my mother
, 905
"... Abstract: The aim of this paper is to try to establish a generic model for the problem that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current sit ..."
Abstract
 Add to MetaCart
Abstract: The aim of this paper is to try to establish a generic model for the problem that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current situation–from Euclid’s second theorem to GreenTao theorem. We analyzed some equivalent necessary conditions that irreducible univariable polynomials with integral coefficients represent infinitely many primes, found new necessary conditions which perhaps imply that there are only finitely many Fermat primes, generalized Euler’s function, the primecounting function and SchinzelSierpinski’s Conjecture and so on, obtained an analogy of the Chinese Remainder Theorem. By proposed obtrusively several conjectures, we gave a new way for determining the existence of some special kinds of primes. Finally, we proposed sufficient and necessary conditions that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. Nevertheless, this is only a beginning and it miles to go. We hope that number theorists consider further it. Keywords: Euclid’s second theorem, Chinese Remainder Theorem,