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A note on factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1

Michal Křížek, Jan Chleboun (1994)

Mathematica Bohemica

We show that any factorization of any composite Fermat number F m = 2 2 m + 1 into two nontrivial factors can be expressed in the form F m = ( k 2 n + 1 ) ( ℓ 2 n + 1 ) for some odd k and ℓ , k ≥ 3 , ℓ ≥ 3 , and integer n ≥ m + 2 , 3 n < 2 m . We prove that the greatest common divisor of k and ℓ is 1, k + ℓ ≡ 0 m o d 2 n , m a x ( k , ℓ ) ≥ F m - 2 , and either 3 | k or 3 | ℓ , i.e., 3 h 2 m + 2 + 1 | F m for an integer h ≥ 1 . Factorizations of F m into more than two factors are investigated as well. In particular, we prove that if F m = ( k 2 n + 1 ) 2 ( ℓ 2 j + 1 ) then j = n + 1 , 3 | ℓ and 5 | ℓ .

A polynomial reduction algorithm

Henri Cohen, Francisco Diaz Y Diaz (1991)

Journal de théorie des nombres de Bordeaux

The algorithm described in this paper is a practical approach to the problem of giving, for each number field K a polynomial, as canonical as possible, a root of which is a primitive element of the extension K / ℚ . Our algorithm uses the L L L algorithm to find a basis of minimal vectors for the lattice of ℝ n determined by the integers of K under the canonical map.

A subresultant theory of multivariate polynomials.

Laureano González Vega (1990)

Extracta Mathematicae

In Computer Algebra, Subresultant Theory provides a powerful method to construct algorithms solving problems for polynomials in one variable in an optimal way. So, using this method we can compute the greatest common divisor of two polynomials in one variable with integer coefficients avoiding the exponential growth of the coefficients that will appear if we use the Euclidean Algorithm.In this note, generalizing a forgotten construction appearing in [Hab], we extend the Subresultant Theory to the...

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