The Hardy–Ramanujan Theorem on the Number of Distinct Prime Divisors

2014 ◽  
pp. 81-87
Author(s):  
Ross G. Pinsky
Keyword(s):  
Prime Divisors

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

scholarly journals The diminished base locus is not always closed

2014 ◽  
Vol 150 (10) ◽  
pp. 1729-1741 ◽  
Author(s):  
John Lesieutre
Keyword(s):  
Blow Up ◽  
Weak Sense ◽  
Prime Divisors ◽  
General Fiber ◽  
Base Locus ◽  
The Family

AbstractWe exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

scholarly journals On multiple prime divisors of cyclotomic polynomials

1974 ◽  
Vol 28 (127) ◽  
pp. 847-847 ◽  
Author(s):  
Wayne L. McDaniel
Keyword(s):  
Cyclotomic Polynomials ◽  
Prime Divisors

scholarly journals Gorenstein Resolutions of 3-Dimensional Terminal Singularities

2005 ◽  
Vol 178 ◽  
pp. 63-115 ◽  
Author(s):  
Takayuki Hayakawa
Keyword(s):  
Minimal Resolution ◽  
Prime Divisors ◽  
Terminal Singularity ◽  
3 Dimensional ◽  
Index 2 ◽  
General Member

Let X be a 3-dimensional terminal singularity of index ≥ 2. We shall construct projective birational morphisms ƒ: Y → X such that Y has only Gorenstein terminal singularities and that ƒ factors the minimal resolution of a general member of | −KX |. We also study prime divisors of ƒ, especially the discrepancies of these prime divisors.

Prime Divisors ofp-regular Conjugacy Class Sizes

2015 ◽  
Vol 43 (8) ◽  
pp. 3365-3371 ◽  
Author(s):  
Yang Liu ◽  
Ziqun Lu
Keyword(s):  
Conjugacy Class ◽  
Conjugacy Class Sizes ◽  
Prime Divisors

Pseudoperfect numbers with no small prime divisors

2009 ◽  
Vol 93 (528) ◽  
pp. 404-409
Author(s):  
Peter Shiu
Keyword(s):  
Difficult Problem ◽  
Prime Divisors ◽  
Perfect Number ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

A perfect number is a number which is the sum of all its divisors except itself, the smallest such number being 6. By results due to Euclid and Euler, all the even perfect numbers are of the form 2P-1(2p - 1) where p and 2p - 1 are primes; the latter one is called a Mersenne prime. Whether there are infinitely many Mersenne primes is a notoriously difficult problem, as is the problem of whether there is an odd perfect number.

scholarly journals A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences

2019 ◽  
Vol 194 ◽  
pp. 8-29 ◽  
Author(s):  
M. Ram Murty ◽  
François Séguin ◽  
Cameron L. Stewart
Keyword(s):  
Lower Bound ◽  
Prime Divisors ◽  
Recurrence Sequences

scholarly journals Prime divisors of some shifted products

2005 ◽  
Vol 2005 (19) ◽  
pp. 3057-3073
Author(s):  
Eric Levieil ◽  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Prime Divisors ◽  
Positive Integers

We study prime divisors of various sequences of positive integersA(n)+1,n=1,…,N, such that the ratiosa(n)=A(n)/A(n−1)have some number-theoretic or combinatorial meaning. In the casea(n)=n, we obviously haveA(n)=n!, for which several new results about prime divisors ofn!+1have recently been obtained.

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