منابع مشابه
The Factorization of the Ninth Fermat Number
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...
متن کاملFactorization of the tenth Fermat number
We describe the complete factorization of the tenth Fermat number F10 by the elliptic curve method (ECM). F10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations
متن کاملFactorization of the Eighth Fermât Number
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.
متن کاملThe Fermat factorization method revisited
We consider the well known Fermat factorization method, we call the Fermat factorization equation the equation solved by it: P(x, y) = (x+ 2R) − y − 4N = 0; where N = p q > 0 is a RSA modulus with primes p and q supposed of equal length. This equation is a bivariate integer polynomial equation and we propose to solve it directly using Coppersmith’s methods for bivariate integer polynomials. As ...
متن کاملThe Twenty-second Fermat Number Is Composite
We have shown by machine proof that F22 = 22 +1 is composite. In addition, we reenacted Young and Buell's 1988 resolution of F20 as composite, finding agreement with their final Selfridge-Hurwitz residues. We also resolved the character of all extant cofactors of Fn , n < 22, finding no new primes, and ruling out prime powers.
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1981
ISSN: 0025-5718
DOI: 10.2307/2007666