All Questions
Tagged with combinatorics number-theory
1,985
questions
-1
votes
1
answer
70
views
Non negative integral solutions when coefficients are involved and inequality among parameters
What is the number of non negative integral solutions of
$5w+3x+y+z=100$, such that $w≤x≤y≤z$.
I have the answer key to this but I do not understand how to even start. The second inequality condition ...
- 183
8
votes
1
answer
263
views
Existence of a Subset $S$ of $\mathbb{N}$ Where All but Finitely Many Natural Numbers Are Sums of Consecutive Elements of $S$
I am pondering a question in number theory that touches upon the representation of natural numbers as sums of consecutive elements from a subset S of $\mathbb{N}$. Specifically, the question is:
Does ...
- 702
1
vote
0
answers
322
views
A conjecture on representing $\sum\limits_{k=0} ^m (-1)^ka^{m-k}b^k$ as sum of powers of $(a+b)$.
UPATE: I asked this question on MO here.
I was solving problem 1.2.52 in "An introduction to the theory of numbers by by Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery"
Show that if ...
- 6,565
1
vote
1
answer
66
views
Sum-free sequence but multiset
The question is:
Show that if $S$ is a set of natural numbers such that no number in S can be expressed as a sum of other (not necessarily distinct) numbers in S, then $\sum_{ s \in S} \frac{1}{s} \...
- 506
0
votes
0
answers
48
views
Find generating series on set of descending sequences, with weight function as taking sum of sequence
Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum ...
- 183
0
votes
0
answers
65
views
Given an integer $n>0$, $\forall 0<k\le n$, the total factor $2$ does the sequence contain?
For example,
if $n=2$, we have a sequence: $1,2$, where $2$ has one factor $2$, so the counting function: $\phi(2)=1$
if $n=4$, we have a sequence: $1,2,3,4$, where $4=2\cdot2$ has two factor $2$, so ...
- 21.2k
1
vote
0
answers
112
views
$q$-Pochhammer at root of unity
Are there any identities, papers/studies, posts, etc that go over
$$(\ln\zeta_n^k;q)_{\infty} = \prod_{m=0}^{\infty}(1-\frac{2\pi i k q^m}{n})$$
which is sometimes called the $q$-Pochhammer or quantum ...
- 702
2
votes
1
answer
104
views
power of a matrix that has a very "big" entry on a column
Given a matrix $A = (a_{i,j}) \in \mathbb{Z}^{n,n} $. We say the entry $a_{r,k} $ on the $k$-th column is big if $a_{r,k} > \frac{1}{2} \sum_{i=1}^n\left|a_{i,k}\right| + 1$ (so that $|a_{r,k}|$ ...
- 1,641
0
votes
0
answers
50
views
Higher dimensional polygonal numbers formula
The general formula for the $m$-th $n$-gonal number is given by
$$P_n(m) = \frac{m^2(n-2)-m(n-4)}{2}$$
So, to give quick examples:
Triangular numbers ($n=3$):
$$P_3(m) = \frac{m^2+m}{2}$$
Square ...
- 702
5
votes
1
answer
205
views
Conjecture: $\binom{n}{k } \mod m =0$ for all $k=1,2,3,\dots,n-1$ only when $m $ is a prime number and $n$ is a power of $m$
While playing with Pascal's triangle, I observed that $\binom{4}{k } \mod 2 =0$ for $k=1,2,3$,and $\binom{8}{k } \mod 2 =0$ for $k=1,2,3,4,5,6,7$ This made me curious about the values of $n>1$ and ...
- 6,565
0
votes
1
answer
44
views
The Asymptotic formula of the generating function related with the partition of a positive integer
This question may be duplicate with this answer_1 and here I referred to the same paper by Hardy, G. H.; Ramanujan, S. referred to by wikipedia which is referred to in answer_1.
But here I focused on ...
- 416
2
votes
0
answers
39
views
AKS primality test lower bound
In this paper, in the first part of Section 7, D.J. Bernstein mentions that the original AKS paper (Lemma 4.4 and its proof) uses an extremely crude lower bound for the number of elements in $G$ (as ...
- 21
2
votes
1
answer
61
views
Counting gap sizes in a subfamily of partitions
Let $\mathcal{OD}$ be the set of all odd and distinct integer partitions. This has a generating function given by
$$\sum_{\lambda\vdash\mathcal{OD}}q^{\vert\lambda\vert}=\prod_{j\geq1}(1+q^{2j-1})$$
...
- 339
4
votes
0
answers
109
views
How many connected nonisomorphic graphs of N vertices given certain edge constraints?
Background:
I’m helping a colleague with a theoretical problem in ecology, and I haven’t quite the background to solve this myself. However, I can state the problem clearly, I think:
Problem statement:...
- 341
2
votes
1
answer
97
views
Show that $n$ has $2^{\omega(n) - 1}$ coprime factor pairs
I am trying to show that $n$ has $2^{\omega(n) - 1}$ coprime factor pairs. I'm pretty sure this is true but I don't see how to prove it. There is no obvious way to use induction.
Here is an example:
$...
- 3,187
2
votes
1
answer
221
views
Generalization of binomial coefficients to both non-integer arguments
It is known that binomial coefficients can be generalized to the following: for $s\in\mathbb R$ and $k\in\mathbb N$,
\begin{equation*}
\binom{s}{k} := \prod_{i=0}^{k-1} \frac{s-i}{k-i} = \frac{s(s-1)...
- 1,972
1
vote
0
answers
24
views
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
0
votes
0
answers
36
views
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&...
- 191
1
vote
0
answers
58
views
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
votes
0
answers
48
views
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
votes
0
answers
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 ...
user1232857
3
votes
1
answer
88
views
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
...
- 702
2
votes
1
answer
70
views
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} \,...
- 89
1
vote
1
answer
47
views
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$ ...
- 848
6
votes
1
answer
300
views
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 ...
- 1,641
1
vote
0
answers
66
views
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
1
vote
0
answers
24
views
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
5
votes
0
answers
77
views
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 ...
0
votes
1
answer
64
views
For Infinites $A,B\subset\mathbb{N}$ s.t exists $n \in A$ and $r,l < n$ s.t the sets $B\cap(n\mathbb{N}+r)$ and $B\cap(n\mathbb{N}+l)$ are infinite.
We note that for every subset $B$ of $\mathbb{N}$ and for every $n \in \mathbb{N}$ we have the following
$$
B = \bigcup_{0 \leq r < n} (n\mathbb{N} + r) \cap B,
$$
where $n\mathbb{N} + r = \{ n \...
- 133