Related papers: Simerka - Quadratic Forms and Factorization

The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced…

We recall Borcherds's approach to vertex algebras via "singular commutative rings", and introduce new examples of his constructions which we compare to vertex algebras, chiral algebras, and factorization algebras. We show that all vertex…

We obtain an explicit formula for the spherical image of the polynomial fraction conjectured by Shimura in 1963 for generating Hecke series in particular case of genus 4. As in our previous work we used the Satake spherical map for ${\rm…

Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using…

In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq…

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…

The linearization of a quadratic form gives rise to a Clifford algebra structure, as seen in Dirac's factorization of the d'Alembert operator. A similar structure known as a generalized Clifford algebra arises from the continuation of this…

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present…

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (Math Comp, 1979, 33: 1293--1297) and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme…

Carmichael quotients for an integer $m\ge 2$ are introduced analogous to Fermat quotients, by using Carmichael function $\lambda(m)$. Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences…

We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat's method in…

We prove that there exist infinitely many (-1,1)-Carmichael numbers, that is, square-free, composite integers n such that p+1 divides n-1 for each prime p dividing n.

We study the problem of decomposition (non-commutative factorization) of linear ordinary differential operators near an irregular singular point. The solution (given in terms of the Newton diagram and the respective characteristic numbers)…

Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the…

Marked mesh patterns are a very general type of permutation pattern. We examine a particular marked mesh pattern originally defined by Kitaev and Remmel, and show that its generating function is described by the $r$-Stirling numbers. We…

Using factorization homology techniques, we give a simple proof of the action of Kontsevich and Soibelman operad on the pair consisting of Hochschild cohomology and Hochschild homology. The proof obviously extends to higher Hochschild…

This paper presents both a method and a result. The result presents a closed formula for the sum of the first $m+1,m \ge 0,$ squares of the sequence $F^{(k)}$ where each member is the sum of the previous $k$ members and with initial…

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same…

An extension to the factorisation principle as suggested by Fermat is presented.We start from a symmetry of natural numbers and obtain the factorisation principle therefrom.Later it is extended further to test the primality of any natural…

We give some new formulas about factorizations of $K$-$k$-Schur functions $g^{(k)}_{\lambda}$, analogous to the $k$-rectangle factorization formula $s^{(k)}_{R_t\cup\lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}$ of $k$-Schur functions, where…