Related papers: A characterization of the Frobenius problem and it…

For an integer $b\geq 2$, a positive integer is called a $b$-Niven number if it is a multiple of the sum of the digits in its base-$b$ representation. In this article, we show that every arithmetic progression contains infinitely many…

We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the…

We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy,…

Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of…

Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron Frobenius-like theory for these matrices, obtaining three main results and drawing…

An apriori bound for the condition number associated to each of the following problems is given: general linear equation solving, minimum squares, non-symmetric eigenvalue problems, solving univariate polynomials, solving systems of…

In a prequel to this paper \cite{curvature1} a notion of curvature on the integers was introduced, based on the technique of "analytic continuation between primes", introduced in \cite{laplace}. In this paper, which is essentially…

A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible…

Given positive integers $n$ and $k$, a $k$-term semi-progression of scope $m$ is a sequence $(x_1,x_2,...,x_k)$ such that $x_{j+1} - x_j \in \{d,2d,\ldots,md\}, 1 \le j \le k-1$, for some positive integer $d$. Thus an arithmetic progression…

Recently, Harrington, Litman, and Wong [Bulletin of the Australian Mathematical Society, 2024; arXiv:2303.06534] proved that every arithmetic progression contains infinitely many base-$b$ Niven numbers, for any fixed $b\ge 2$. We use a…

The Unbounded Subset Sum (USS) problem is an NP-hard computational problem where the goal is to decide whether there exist non-negative integers $x_1, \ldots, x_n$ such that $x_1 a_1 + \ldots + x_n a_n = b$, where $a_1 < \cdots < a_n < b$…

For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied…

The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we…

Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

The Frobenius method can be used to represent solutions of ordinary differential equations by (generalized) power series. It is useful to have prior knowledge of the coefficients of this series. In this contribution we demonstrate that the…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

Let $\varphi(x_1,\ldots,x_h,y) = u_1x_1 + \cdots + u_hx_h+vy$ be a linear form with nonzero integer coefficients $u_1,\ldots, u_h, v.$ Let $\mathcal{A} = (A_1,\ldots, A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an…

The Frobenius number $g(S)$ of a set $S$ of non-negative integers with $\gcd 1$ is the largest integer not expressible as a linear combination of elements of $S$. Given a sequence ${\bf s} = (s_i)_{i \geq 0}$, we can define the associated…

Frobenius algebras in the category of sets and relations ($\mathbf{Rel}$) serve as a unifying framework for various algebraic and combinatorial structures, including groupoids, effect algebras, and abstract circles. Recently, a nerve…