All Questions
9 questions
1
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1
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186
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Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
4
votes
1
answer
258
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Density of numbers where a large prime factor satisfies a congruence
I am looking for an upper bound on the number of integers $n<x$ such that $n$ has a prime factor $p>\log(x)^{(1+\delta)}$ such that $p \equiv a \mod b$. Where $a,b$ are fixed and coprime and $0&...
0
votes
0
answers
102
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On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...
- 13.9k
4
votes
1
answer
336
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Infinite, finitely generated linear group has indices of subgroups divisible by infinitely many primes
This question was previously asked at Math.SE, but didn't receive much attention.
Let $G$ be an infinite finitely generated group and suppose that $G$ is linear, say $G \leq \operatorname{GL}_n(K)$ ...
- 2,821
8
votes
1
answer
575
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Unstable Integers
There is a question that has been bothering my mind for quite a while now. I will present it and my current thoughts and progress on it.
Let the prime factorization of an integer $n$ be
$$n = p_1^{...
- 311
5
votes
1
answer
499
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What is the asymptotic growth of $\sum_{k=1}^n 2^{\omega_k}$?
Question: Let $\omega_k$ be the number of distinct prime divisors of k.
What is the asymptotic growth of $C_n := \sum_{k=1}^n 2^{\omega_k}$?
Thank you for considering this elementary question. ...
- 2,374
4
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0
answers
199
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Linear combination of characters
For each $i \in \mathbb{N}$, let $G_{i}$ be a finite abelian group and $\widehat{G_{i}}$ the $\overline{\mathbb{Q}}$-valued character group of $G_{i}$. Suppose that $|G_{i}| \rightarrow \infty$ as $i \...
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1
vote
2
answers
322
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group theory behind the Kloosterman bound $| S(m,n;c) |< 2\, c^{3/4}$
I am trying to understand Kloosterman sums and their estimates (e.g. from [1], which does not prove)
$$ \Big| S(m,n;c) \Big| = \Big| \sum_{(x,c) = 1} e\big( \frac{mx + nx^{-1}}{c}\big) \Big| < 2\, ...
- 22.8k
7
votes
3
answers
582
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Asymptotics for the number of abelian groups of order at most $x.$
The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for $a(...
- 96.4k