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Beta expansion of Salem numbers approaching Pisot numbers with the finiteness property

Hachem Hichri (2015)

Acta Arithmetica

It is already known that all Pisot numbers are beta numbers, but for Salem numbers this was proved just for the degree 4 case. In 1945, R. Salem showed that for any Pisot number θ we can construct a sequence of Salem numbers which converge to θ. In this short note, we give some results on the beta expansion for infinitely many sequences of Salem numbers obtained by this construction.

Census algorithms for chinese remainder pseudorank

David Laing, Bruce Litow (2007)

RAIRO - Theoretical Informatics and Applications

We investigate the density and distribution behaviors of the chinese remainder representation pseudorank. We give a very strong approximation to density, and derive two efficient algorithms to carry out an exact count (census) of the bad pseudorank integers. One of these algorithms has been implemented, giving results in excellent agreement with our density analysis out to 5189-bit integers.

Census algorithms for chinese remainder pseudorank

David Laing, Bruce Litow (2008)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We investigate the density and distribution behaviors of the chinese remainder representation pseudorank. We give a very strong approximation to density, and derive two efficient algorithms to carry out an exact count (census) of the bad pseudorank integers. One of these algorithms has been implemented, giving results in excellent agreement with our density analysis out to 5189 -bit integers.

Denominators of Igusa class polynomials

Kristin Lauter, Bianca Viray (2014)

Publications mathématiques de Besançon

In [22], the authors proved an explicit formula for the arithmetic intersection number CM ( K ) . G 1 on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field K . These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross...

Extended Euclidean Algorithm and CRT Algorithm

Hiroyuki Okazaki, Yosiki Aoki, Yasunari Shidama (2012)

Formalized Mathematics

In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based...

GiANT: Graphical Algebraic Number Theory

Aneesh Karve, Sebastian Pauli (2006)

Journal de Théorie des Nombres de Bordeaux

While most algebra is done by writing text and formulas, diagrams have always been used to present structural information clearly and concisely. Text shells are the de facto interface for computational algebraic number theory, but they are incapable of presenting structural information graphically. We present GiANT, a newly developed graphical interface for working with number fields. GiANT offers interactive diagrams, drag-and-drop functionality, and typeset formulas.

Hierarchical residue number systems with small moduli and simple converters

Tadeusz Tomczak (2011)

International Journal of Applied Mathematics and Computer Science

In this paper, a new class of Hierarchical Residue Number Systems (HRNSs) is proposed, where the numbers are represented as a set of residues modulo factors of 2k ± 1 and modulo 2k . The converters between the proposed HRNS and the positional binary number system can be built as 2-level structures using efficient circuits designed for the RNS (2k-1, 2k, 2k+1). This approach allows using many small moduli in arithmetic channels without large conversion overhead. The advantages resulting from the...

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function ψ ( x , y ) which is the number of positive integers x and free of prime factors > y . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( n 3 ) real right-angled simplices. In this paper, we...

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