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pro vyhledávání: '"A, Pomerance"'
Autor:
Pomerance, Carl
Mersenne primes and Fermat primes may be thought of as primes of the form $\Phi_m(2)$, where $\Phi_m(x)$ is the $m$th cyclotomic polynomial. This paper discusses the more general problem of primes and composites of this form.
Comment: 12 pages
Comment: 12 pages
Externí odkaz:
http://arxiv.org/abs/2411.04213
Autor:
Fan, Steve, Pomerance, Carl
Let $\omega^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of $\omega^*(n)
Externí odkaz:
http://arxiv.org/abs/2401.10427
Autor:
Fan, Steve, Pomerance, Carl
Let $\Phi(x,y)$ denote the number of integers $n\in[1,x]$ free of prime factors $\le y$. We show that but for a few small cases, $\Phi(x,y)<.6x/\log y$ when $y\le\sqrt{x}$.
Comment: 15 pages, 1 figure; to appear in the Journal of Number Theory
Comment: 15 pages, 1 figure; to appear in the Journal of Number Theory
Externí odkaz:
http://arxiv.org/abs/2306.03339
Autor:
Wikner, Alexander, Harvey, Joseph, Girvan, Michelle, Hunt, Brian R., Pomerance, Andrew, Antonsen, Thomas, Ott, Edward
Recent work has shown that machine learning (ML) models can be trained to accurately forecast the dynamics of unknown chaotic dynamical systems. Short-term predictions of the state evolution and long-term predictions of the statistical patterns of th
Externí odkaz:
http://arxiv.org/abs/2211.05262
Autor:
Albright, Connor, Hadaway, Kimberly P., Pomerance, Ari Holcombe, Jeffries, Joel, Lorenzen, Kate J., Nix, Abigail K.
In a connected graph, Kemeny's constant gives the expected time of a random walk from an arbitrary vertex $x$ to reach a randomly-chosen vertex $y$. Because of this, Kemeny's constant can be interpreted as a measure of how well a graph is connected.
Externí odkaz:
http://arxiv.org/abs/2211.01495
We introduce a mathematical framework for simulating Hybrid Boolean Network (HBN) Physically Unclonable Functions (PUFs, HBN-PUFs). We verify that the model is able to reproduce the experimentally observed PUF statistics for uniqueness $\mu_{inter}$
Externí odkaz:
http://arxiv.org/abs/2207.10816
Autor:
Pomerance, Carl
Let $S_{\rm lcm}(n)$ denote the set of permutations $\pi$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,\pi(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $\pi$ of $[n]$ such that $j\mid\pi(j)$ or $\pi
Externí odkaz:
http://arxiv.org/abs/2206.01699
Autor:
Pomerance, Carl
Let $C(n)$ denote the number of permutations $\sigma$ of $[n]=\{1,2,\dots,n\}$ such that $\gcd(j,\sigma(j))=1$ for each $j\in[n]$. We prove that for $n$ sufficiently large, $n!/3.73^n < C(n) < n!/2.5^n$.
Externí odkaz:
http://arxiv.org/abs/2203.03085
Autor:
Pomerance, Carl
We prove that there is a matching between 2 intervals of positive integers of the same even length, with corresponding pairs coprime, provided the intervals are in $[n]$ and their lengths are $>c(\log n)^2$, for a positive constant $c$. This improves
Externí odkaz:
http://arxiv.org/abs/2111.07157
Recent research has established the effectiveness of machine learning for data-driven prediction of the future evolution of unknown dynamical systems, including chaotic systems. However, these approaches require large amounts of measured time series
Externí odkaz:
http://arxiv.org/abs/2110.03722