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pro vyhledávání: '"Pomerance, Carl"'
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:
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
Autor:
Luca, Florian, Pomerance, Carl
It is conjectured that the sum $$ S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k} $$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$
Externí odkaz:
http://arxiv.org/abs/2106.08275
We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct Bernoulli denominators and some related problems.
Comment: 15 pages. Some references added,
Comment: 15 pages. Some references added,
Externí odkaz:
http://arxiv.org/abs/2105.13252
We consider several problems about pseudoprimes. First, we look at the issue of their distribution in residue classes. There is a literature on this topic in the case that the residue class is coprime to the modulus. Here we provide some robust stati
Externí odkaz:
http://arxiv.org/abs/2103.00679
Autor:
Pomerance, Carl
Improving on some recent results of Matom\"aki and of Wright, we show that the number of Carmichael numbers to $X$ in a coprime residue class exceeds $X^{1/(6\log\log\log X)}$ for all sufficiently large $X$ depending on the modulus of the residue cla
Externí odkaz:
http://arxiv.org/abs/2101.09906