All Questions
Tagged with combinatorics number-theory
1,985
questions
1
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2
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325
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Lower bound related to Goldbach conjecture
I am curious to know if a lower bound on the number of ways (call this $\beta$ and assume $p_1 + p_2$ distinct from $p_2 + p_1$) in which two primes $p_1, p_2$ that add up to a given even integer $n$, ...
- 10.3k
1
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0
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24
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Notation for $k$-partitions of $n$ containing at least one summand equal to $s$
I am looking for whether there is any notation for the $k$-partition number of $n$ where the partitions must include some summand $s$.
An example of the kind of notation I am looking for is $P_k^s(n)$....
- 1,129
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0
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36
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Show explicitly the rank-24 Leech lattice that is also symmetric unimodular matrix with integer entries?
A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix:
$$\begin{pmatrix}
2 &−1 &0 &0& 0& 0& 0& 0\\
−1& 2 &−1& 0& 0& 0& 0& 0\\
0& −1&...
- 1,216
1
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0
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58
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Exploring a Novel Pattern in Exponential Number Sequences and Their Relation to Factorials
Introduction
I've been exploring a concept in mathematics that I've termed "string math," which involves examining patterns in exponential number sequences and their intriguing connection to ...
- 11
0
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0
answers
48
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Optimal Strategy for Identifying Lighter Balls: A Balance Scale Puzzle
There are n balls, among which m balls are lighter (and equally light with each other). We have a balance scale; how many times must we weigh at least, in order to find these m lighter balls? We ...
0
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0
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38
views
Schnirelmann density and bases of finite order
Let $\mathcal{A}$ be an additive set. We know that if the Schnirelmann density $\sigma_{\mathcal{A}}$ is positive then it is a basis of finite order. But it it not a necessary condition. My question ...
CommunityBot
- 1
12
votes
3
answers
19k
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Calculating integer partitions
A partition of a positive integer $n$, also called an integer partition, is a way of writing $n$ as a sum of positive integers. The number of partitions of $n$ is given by the partition function $p(n)$...
- 47.4k
3
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1
answer
88
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Is there a smallest large set?
A set $A = \{a_1, a_2 ,..\}$ of positive integers is called large if $\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + ...$ diverges. A small set is any set of the positive integers that is not large
...
- 2,753
2
votes
1
answer
70
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Identity involving Sum of Inverse of Product of Integer partitions [closed]
Is there a way to prove the following identity
\begin{equation}
\sum_{l = 1}^{k} \left( \frac{(-s)^l}{l!} \sum_{n_1 + n_2 + \ldots n_l = k} \frac{1}{n_1n_2 \ldots n_k} \right)= (-1)^k {s \choose k} \,...
- 907
6
votes
1
answer
300
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Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I am new to number theory, I was wondering if the following questions have been studied before.
Given $f(x) = a_0 + a_1 x + a_2 x^2 \cdots + a_n x^n \in \mathbb Z[x]$, we say that $f(x)$ has a big ...
- 9,080
1
vote
1
answer
47
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When is it possible to partition $n(n-1)/2$ pairs of $\{i,j\}$ into $n-1$ sets such that in each set each number appears once?
Consider an even number $n$. There are $\frac{n(n-1)}{2}$ pairs of $\{i,j\}$ with $1 \leq i < j \leq n$. Consider the following problem. The goal is to allocate each pair of $\{i,j\}$ to $n-1$ ...
- 71
1
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0
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66
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Lower bound for a counting function of products of primes
Let $P$ be a non-empty finite set of prime numbers, and let $S(P)$ be such that $(\rm i)$ $P \subset S(P)$ and $(\rm ii)$ $p q \in S(P)$ for every $p, q \in S(P)$. Hence, $S(P)$ is the set of all ...
- 179
2
votes
0
answers
51
views
Number of divisor sequences of a number
Define a sequence, $(a_k)_{k=1}^m$ to be a divisor sequence of a positive integer $n$ if it satisfies the following:
$a_1=1$
$a_m=n$
$a_{k-1} \mid a_k$ and $0<a_{k-1}<a_{k}$ for all $1<k\le ...
- 21
5
votes
0
answers
77
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Number of loops of a ball bouncing in a room with obstacles
Introduction
With a friend of mine we were studying the following problem: given a $m\times n$ grid draw this pattern (I don't know how to describe it in words)
The first image has $3$ loops and the ...
1
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0
answers
24
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Sum of the restrictions of Z ideals to an interval
I am currently studying a combinatorics question that makes appear the following type of sets:
$p\mathbb{Z}\cap [n]+q\mathbb{Z}\cap[n]$.
It is basically interescting ideals of $\mathbb{Z}$, but ...
- 11