A problem involving integers all of whose prime divisors belong to given arithmetic progressions

2016 ◽  
Vol 71 (4) ◽  
pp. 790-792 ◽  
Author(s):  
M E Changa
Keyword(s):  
Arithmetic Progressions ◽  
Prime Divisors

On the maximal length of two sequences of integers in arithmetic progressions with the same prime divisors

1996 ◽  
Vol 121 (4) ◽  
pp. 295-307 ◽  
Author(s):  
R. Balasubramanian ◽  
M. Langevin ◽  
T. N. Shorey ◽  
M. Waldschmidt
Keyword(s):  
Arithmetic Progressions ◽  
Maximal Length ◽  
Prime Divisors

scholarly journals Erdős-Ginzburg-Ziv Constants by Avoiding Three-Term Arithmetic Progressions

10.37236/7275 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Jacob Fox ◽  
Lisa Sauermann
Keyword(s):  
Abelian Group ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Upper Bounds ◽  
Arithmetic Progressions ◽  
Direct Connection ◽  
Prime Divisors ◽  
Zero Sum ◽  
Special Case

For a finite abelian group $G$, The Erdős-Ginzburg-Ziv constant $\mathfrak{s}(G)$ is the smallest $s$ such that every sequence of $s$ (not necessarily distinct) elements of $G$ has a zero-sum subsequence of length $\operatorname{exp}(G)$. For a prime $p$, let $r(\mathbb{F}_p^n)$ denote the size of the largest subset of $\mathbb{F}_p^n$ without a three-term arithmetic progression. Although similar methods have been used to study $\mathfrak{s}(G)$ and $r(\mathbb{F}_p^n)$, no direct connection between these quantities has previously been established. We give an upper bound for $\mathfrak{s}(G)$ in terms of $r(\mathbb{F}_p^n)$ for the prime divisors $p$ of $\operatorname{exp}(G)$. For the special case $G=\mathbb{F}_p^n$, we prove $\mathfrak{s}(\mathbb{F}_p^n)\leq 2p\cdot r(\mathbb{F}_p^n)$. Using the upper bounds for $r(\mathbb{F}_p^n)$ of Ellenberg and Gijswijt, this result improves the previously best known upper bounds for $\mathfrak{s}(\mathbb{F}_p^n)$ given by Naslund. 

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 Colorings with only rainbow arithmetic progressions

2020 ◽  
Vol 161 (2) ◽  
pp. 507-515
Author(s):  
J. Pach ◽  
I. Tomon
Keyword(s):  
Arithmetic Progressions

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.

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