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On the Number of Primality Witnesses of Composite Integers
Russian Mathematics, 2021The Miller-Rabin test tries to determine the primality of a given odd number \(n\), see [\textit{M. O. Rabin}, J. Number Theory 12, 128--138 (1980; Zbl 0426.10006)].
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Compositions of Positive Integers
The Mathematics Teacher, 1991During the summer of 1988, the first author conducted an institute on discrete mathematics for sixteen New Mexico high school teachers. One of the presented topics, compositions of integers, proved to be a fruitful source of interesting problems.
Richard M. Grassl +2 more
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On the compositions of an integer
1980We prove that for all positive integers n, the number of compositions of n in which the largest part is m is a unimodal function of m.
A. Odlyzko, B. Richmond
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Longest Run of Equal Parts in a Random Integer Composition
Discrete Mathematics, 2015 15 pages, 3 ...
Ayla Gafni
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On the Mean Value of the Index of Composition of an Integer
Monatshefte für Mathematik, 2005For each integer \(n\geq 2\), let \(\lambda(n)= \frac{\log n}{\log\gamma(n)}\) be the index of composition of \(n\), where \(\gamma(n)= \prod_{p|n}p\). For convenience, we write \(\lambda(1)= \gamma(1)= 1\). The index of composition of an integer measures essentially the multiplicity of its prime factors. It was shown by \textit{J. M.
De Koninck, J. M., Kátai, I.
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Distinctness of compositions of an integer: A probabilistic analysis
Random Structures & Algorithms, 2001AbstractCompositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. In this article, we use as measure of distinctness the number of distinct parts (or components).
Pawel Hitczenko, Guy Louchard
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A New Method for Generating Integer Compositions in Parallel
International Journal of Parallel, Emergent and Distributed Systems, 1996A novel method, binary countdown transformation, for generating integer compositions in parallel is proposed. Based on this method, an optimal and highly adaptive parallel algorithm for generating all compositions of integer n is presented. The algorithm runs on p processors on a simple SIMD model requiring neither shared memory nor interconnection ...
David J Evans
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Counting staircases in integer compositions
Online Journal of Analytic Combinatorics, 2016The main theorem establishes the generating function \(F\) which counts the number of times the staircase \(1 + 2 + 3 + \cdots + m^+\) fits inside an integer composition of \(n\). \[ F = \frac{k_m - \frac{q x^m y}{1-x} k_{m-1}}{(1-q)x^{\binom{m+1}{2}} \left( \frac{y}{1-x} \right)^m + \frac{1-x-xy}{1-x} \left( k_m - \frac{q x^m y}{1-x} k_{m-1} \right)}.
Blecher, Aubrey, Mansour, Toufik
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Integers are only Primes and Composites
2023(Internal Code: C002800) Abstract: This study establishes that integers can exclusively fall into the categories of prime or composite numbers. The rationale behind this arises from the imperative inclusion of (I) negative prime numbers, (II) the acknowledgment of 1 and -1 as prime numbers, and (III) the incorporation of 0 as a composite number within
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The water capacity of integer compositions
Online Journal of Analytic Combinatorics, 2018We introduce the notion of capacity (ability to contain water) for compositions. Initially the compositions are defined on a finite alphabet \([k]\) and thereafter on \(\mathbb{N}\). We find a capacity generating function for all compositions, the average capacity generating function and an asymptotic expression for the average capacity as the size of
Blecher, Aubrey +2 more
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