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Improvements on the Cantor-Zassenhaus factorization algorithm

Michele Elia, Davide Schipani (2015)

Mathematica Bohemica

The paper presents a careful analysis of the Cantor-Zassenhaus polynomial factorization algorithm, thus obtaining tight bounds on the performances, and proposing useful improvements. In particular, a new simplified version of this algorithm is described, which entails a lower computational cost. The key point is to use linear test polynomials, which not only reduce the computational burden, but can also provide good estimates and deterministic bounds of the number of operations needed for factoring....

Increasing integer sequences and Goldbach's conjecture

Mauro Torelli (2006)

RAIRO - Theoretical Informatics and Applications

Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences.

Index form equations in quintic fields

István Gaál, Kálmán Győry (1999)

Acta Arithmetica

The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit...

Influence of modeling structure in probabilistic sequential decision problems

Florent Teichteil-Königsbuch, Patrick Fabiani (2006)

RAIRO - Operations Research

Markov Decision Processes (MDPs) are a classical framework for stochastic sequential decision problems, based on an enumerated state space representation. More compact and structured representations have been proposed: factorization techniques use state variables representations, while decomposition techniques are based on a partition of the state space into sub-regions and take advantage of the resulting structure of the state transition graph. We use a family of probabilistic exploration-like...

Integral points on the elliptic curve y 2 = x 3 - 4 p 2 x

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

Let p be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve E : y 2 = x 3 - 4 p 2 x . Further, let N ( p ) denote the number of pairs of integral points ( x , ± y ) on E with y > 0 . We prove that if p ≥ 17 , then N ( p ) ≤ 4 or 1 depending on whether p ≡ 1 ( mod 8 ) or p ≡ - 1 ( mod 8 ) .

Integrals of logarithmic and hypergeometric functions

Anthony Sofo (2016)

Communications in Mathematics

Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.

Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

Let ℚ ( α ) and ℚ ( β ) be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, ℚ ( β ) → ℚ ( α ) . The algorithm is particularly efficient if there is only one isomorphism.

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