Related papers: Patterns in rational base number systems
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to…
We use representations and differentiation algorithms of posets, in order to obtain results concerning unsolved problems on figurate numbers. In particular, we present criteria for natural numbers which are the sum of three octahedral…
This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the…
This talk describes how a combination of symbolic computation techniques with first-order theorem proving can be used for solving some challenges of automating program analysis, in particular for generating and proving properties about the…
For a prime $p$ and a positive integer $s$ consider a homogeneous linear system over the ring $\mathbb{Z}_{p^s}$ (the ring of integers modulo $p^s$) described by an $n \times m$-matrix. The possible number of solutions to such a system is…
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…
A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n = (a_k a_{k-1} \cdots a_0)_b$ in base $b$ with $a_k \neq 0$ has the symmetric property $a_{k-i} = a_i$ for every…
Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which…
A total set of states for which we have no resolution of the identity (a `pre-basis'), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes…
We describe recurring patterns of numbers that survive each wave of the Sieve of Eratosthenes, including symmetries, uniform subdivisions, and quantifiable, predictive cycles that characterize their distribution across the number line. We…
Let s_q(n) denote the base q sum of digits function, which for n<x, is centered around (q-1)/2 log_q x. In Drmota, Mauduit and Rivat's 2009 paper, they look at sum of digits of prime numbers, and provide asymptotics for the size of the set…
Systems of the form $x = (A x^s)^{1/s} + b$ arise in a range of economic, financial and control problems, where $A$ is a linear operator acting on a space of real-valued functions (or vectors) and $s$ is a nonzero real value. In these…
We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving…
Moebius number systems represent points using sequences of Moebius transformations. Thorough the paper, we are mainly interested in representing the unit circle (which is equivalent to representing R\cup\{\infty\}). The main aim of the…
Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis…
Inspired by a question of Kra, Moreira, Richter, and Robertson, we prove two new results about infinite polynomial configurations in large subsets of the rational numbers. First, given a finite coloring of $\mathbb{Q}$, we show that there…
Norm forms, examples of which include $x^2 + y^2$, $x^2 + x y - 57 y^2$, and $x^3 + 2 y^3 + 4 z^3 - 6 x y z$, are integral forms arising from norms on number fields. We prove that the natural density of the set of integers represented by a…
Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same…
We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms…
We discuss the base 3/2 representation of the natural numbers. We prove that the sum of digits function of the representation is a fixed point of a 2-block substitution on an infinite alphabet, and that this implies that sum of digits…