classes of‎ ‎abaffy-broyden-spedicato (abs) methods have been introduced for‎ ‎solving linear systems of equations‎. ‎the algorithms are powerful methods for developing matrix‎ ‎factorizations and many fundamental numerical linear algebra processes‎. ‎here‎, ‎we show how to apply the abs algorithms to devise algorithms to compute the wz and zw‎ ‎factorizations of a nonsingular matrix as well as...

Ordinary multiplication of natural numbers can be generalized to a ternary operation by considering discrete volumes lattice hexagons. With this operation, notion ‘3-primality’ -primality with respect multiplication- is defined, and it turns out that there are very few 3-primes. They correspond imaginary quadratic fields Q(√-n), n > 0, odd discriminant whose ring integers admits unique facto...

In this paper we exhibit the full prime factorization of the ninth Fermât number Fg = 2512 + 1 . It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two largest prime factors by means of the number field sieve, which is a factoring algorithm that depends on arithmetic in an algebraic number field. In the present case, the number field used was Q(v^2). T...

Factorization algorithms over Q[X] and Fp[X] are key tools of computational number theory. Many algorithms over number fields rely on the possibility of factoring polynomials in those fields. Because of the recent development of relative methods in computational number theory, see for example (Cohen et al. 1998, Daberkow and Pohst 1995), efficient generalizations of factorization algorithms to ...

We describe a Monte Carlo factorization algorithm which was used to factorize the Fermât number F% = 2256 + 1. Previously Fs was known to be composite, but its factors were unknown.