All Questions
8 questions
16
votes
2
answers
1k
views
The Stable Set Conjecture
A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation
$$n\in \mathcal S \iff dn\in \mathcal S$$
holds for almost all positive integers $n$. ...
- 791
1
vote
1
answer
115
views
Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order
Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...
- 3,625
0
votes
1
answer
298
views
Discrepancy in non-homogeneous arithmetic progressions
I have a doubt, Roth's discrepancy theorem [1] says that there is a subset of arithmetic progressions $A \in [n]$ where any function $f:N\rightarrow \left \{ -1,1 \right \} $ implies
$ \left | \...
4
votes
1
answer
314
views
Asymptotic for number of partitions of $n$ into $k$ squares, uniform in $n,k \to +\infty$
Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since
$$9 = 1^2 + 1^2 + 1^2 + ...
- 41
2
votes
1
answer
191
views
Representation numbers of numerical semigroups
I've been playing around with numerical semigroups lately. I'm pretty new to this stuff, so I apologize in advance if my notation is non-standard. Fix positive integers $x_1,\dots,x_r$ with $\gcd(x_1,\...
- 21
23
votes
3
answers
3k
views
How many different numbers can be obtained as product of first $n$ natural numbers?
Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set
$\{1^{a_1} \cdot 2^{...
- 323
1
vote
0
answers
189
views
Estimates for the size of the product set [n].[n] [duplicate]
Possible Duplicate:
Number of elements in the set {1,…,n}*{1,..,n}
Writing $[n]$ for the set $\lbrace1,2,...,n\rbrace$, let $P_n$ denote the product set $[n].[n]$, i.e.
$$ P_n = \lbrace ab ...
- 21
58
votes
3
answers
6k
views
Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$
Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
- 897