Character sums and products of factorials modulo
In this paper, we apply character sum estimates to study products of factorials modulo .
Characterization of power digraphs modulo
A power digraph modulo , denoted by , is a directed graph with as the set of vertices and as the edge set, where and are any positive integers. In this paper we find necessary and sufficient conditions on and such that the digraph has at least one isolated fixed point. We also establish necessary and sufficient conditions on and such that the digraph contains exactly two components. The primality of Fermat number is also discussed.
Characterization of the unique expansions and related problems
Characterizations of Lambek-Carlitz type
We give Lambek-Carlitz type characterization for completely multiplicative reduced incidence functions in Möbius categories of full binomial type. The -analog of the Lambek-Carlitz type characterization of exponential series is also established.
Characterizations of Midy's property.
Characterizing completely multiplicative functions by generalized Möbius functions.
Characterizing Frobenius semigroups by filtration.
Charles Hermite’s stroll through the Galois fields
Although everything seems to oppose the two mathematicians, Charles Hermite’s role was crucial in the study and diffusion of Évariste Galois’s results in France during the second half of the nineteenth century. The present article examines that part of Hermite’s work explicitly linked to Galois, the reduction of modular equations in particular. It shows how Hermite’s mathematical convictions—concerning effectiveness or the unity of algebra, analysis and arithmetic—shaped his interpretation of Galois...
Cipolla pseudoprimes.
Class Number Two for Real Quadratic Fields of Richaud-Degert Type
2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.This paper contains proofs of conjectures made in [16] on class number 2 and what this author has dubbed the Euler-Rabinowitsch polynomial for real quadratic fields. As well, we complete the list of Richaud-Degert types given in [16] and show how the behaviour of the Euler-Rabinowitsch polynomials and certain continued fraction expansions come into play in the complete determination...
Closed form expressions for several Ramanujan sums
Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers
We study some arithmetical and combinatorial properties of β-integers for β being the larger root of the equation x2 = mx - n,m,n ∈ ℵ, m ≥ n +2 ≥ 3. We determine with the accuracy of ± 1 the maximal number of β-fractional positions, which may arise as a result of addition of two β-integers. For the infinite word uβ> coding distances between the consecutive β-integers, we determine precisely also the balance. The word uβ> is the only fixed point of the morphism A → Am-1B and B → Am-n-1B. In...
Combinatorial aspects of the generalized Euler's totient.
Combinatorial aspects of the sequence of Milnor numbers of deformations
We use combinatorics to describe the topology of a singular irreducible plane curve germ f = 0 under small perturbation of parameters.
Combinatorial congruences and Stirling numbers
Combinatorial properties of products of graphs
Comments on the height reducing property
A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus...
Compact integers and factorials
Comparison of algorithms for calculation of the greatest common divisor of several polynomials
The computation of the greatest common divisor (GCD) has many applications in several disciplines including computer graphics, image deblurring problem or computing multiple roots of inexact polynomials. In this paper, Sylvester and Bézout matrices are considered for this purpose. The computation is divided into three stages. A rank revealing method is shortly mentioned in the first one and then the algorithms for calculation of an approximation of GCD are formulated. In the final stage the coefficients...
Comparison of regular convolutions