Related papers: Divisibility of generalized Mersenne numbers

The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation…

Motivated by recently developed interest to the distribution of $q$-ary digits of Mersenne numbers $M_p = 2^p-1$, where $p$ is prime, we estimate rational exponential sums with $M_p$, $p \leq X$, modulo a large power of a fixed odd prime…

In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…

We first introduce the Hamming distance between two strings. Then, we apply this concept to the representations of whole numbers in base n for all positive integers n > 2. We claim that a simple formula exists for the sum of all Hamming…

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial…

We define Macaulay bases of modules, which are a common generalization of Groebner bases and Macaulay $H$-bases to suitably graded modules over a commutative graded $\mathbf{k}$-algebra, where the index sets of the two gradings may differ.…

We study the divisibility of the sums of the odd power of consecutive integers, $S(m,k)=1^{mk}+2^{mk}+\cdots+k^{mk}$ and $1^k+2^k+\cdots+n^k$ for odd integers $m$ and $k$, by using the Girard-Waring identity. Faulhaber's approach for the…

In this paper, we provide a necessary and sufficient condition ensuring the property of exponential dichotomy for periodic linear systems of generalized differential equations. This condition allow us to revisit a recent result of…

We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random…

We improve unconditional estimates on $\Delta_k(x)$, the remainder term of the generalised divisor function, for large $k$. In particular, we show that $\Delta_k(x) \ll x^{1 - 1.889k^{-2/3}}$ for all sufficiently large fixed $k$.

We extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an…

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. Its mean and variance are known, but results for its median and mode are difficult to obtain, although a few cases have been solved and upper/lower…

In arXiv:2212.14023 a decomposition of Gaussian measures on finite-dimensional spaces was introduced, which turned out to be a central technical tool to improve currently known bounds on a long standing conjecture in statistical mechanics…

The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the…

This note examines the infinite divisibility of density-based transformations of normal random variables. We characterize a class of density-based transformations of normal variables which produces non-infinitely divisible distributions. We…

We investigate the number of parts modulo $m$ of $m$-ary partitions of a positive integer $n$. We prove that the number of parts is equidistributed modulo $m$ on a special subset of $m$-ary partitions. As consequences, we explain when the…

Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian…

We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize…

In this paper we prove a generalization of Montel's theorem for a class of first order elliptic equations with measurable coefficients involving Hodge-Dirac operators. We then apply this result to sequences of solutions of second order…

We state some inequalities for m-divisible and infinite divisible characteristic functions. Basing on them we propose a statistical test for a distribution to be infinitely divisible. Keywords: infinite divisible distributions; statistical…