Related papers: Dirichlet law for factorization of integers, polyn…

Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely…

In this paper, we present an approach to integer factorization using distributed representations formed with Vector Symbolic Architectures. The approach formulates integer factorization in a manner such that it can be solved using neural…

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely…

We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our…

An integer $n$ is said to be \textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an…

In this paper, we establish a demi-distributions theory which develops the usual distribution theory, in particular, we show that many conclusions as differentiations, Fourier transforms and convolutions can be generalized to the…

Let $a,b\in \mathbb{N}$ be fixed and coprime such that $a>b$, and let $N$ be any number of the form $a^n\pm b^n$, $n\in\mathbb{N}$. We will generalize a result of Bostan, Gaudry and Schost and prove that we may compute the prime…

In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In…

This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be performed with a first order differential calculus or needs a second order differential calculus.…

We present a polynomial time algorithm, which solves a nonstandard Variation of the well-known PARTITION-problem: Given positive integers $n, k$ and $t$ such that $t \geq n$ and $k \cdot t = {n+1 \choose 2}$, the algorithm partitions the…

In the paper Factorisation of division polynomials (H. Verdure, Proc. japan Academy, Ser A. 80 n. 5), Verdure gives the factorisation patterns of division polynomials of elliptic curves defined over a finite field. However, the result given…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

Dirichlet distribution and Dirichlet process as its infinite dimensional generalization are primarily used conjugate prior of categorical and multinomial distributions in Bayesian statistics. Extensions have been proposed to broaden…

We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of…

We consider the Gaussian limit law for the distribution of the middle prime factor of an integer, defined according to multiplicity or not. We obtain an optimal bound for the speed of convergence, thereby improving on previous estimates…

The problem of a differential operator left- and right division is solved in terms of generalized Bell polinomials for nonabelian differential unitary ring. The definition of the polinomials is made by means of recurrent relations. The…

Elementary proofs of unique factorization in rings of arithmetic functions using a simple variant of Euclid's proof for the fundamental theorem of arithmetic.

In this paper we give a factorization theorem for the ring of exponential polynomials in many variables over an algebraically closed field of characteristic 0 with an exponentiation. This is a generalization of the factorization theorem due…

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…

The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with…