All Questions
Tagged with integer-partitions elementary-number-theory
10
questions
18
votes
6
answers
9k
views
Prime Partition
A prime partition of a number is a set of primes that sum to the number. For instance, {2 3 7} is a prime partition of $12$ because $2 + 3 + 7 = 12$. In fact, there ...
- 587
20
votes
1
answer
4k
views
Partition of ${1, 2, ... , n}$ into subsets with equal sums.
The following is one of the old contest problems (22nd All Soviet Union Math Contest, 1988).
Let $m, n, k$ be positive integers such that $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $...
- 383
4
votes
1
answer
243
views
How to count all the solutions for $\sum\limits_{i=1}^{n} \frac{1}{2^{k_i}}= 1$ for $k_i\in \Bbb{N}$ and $n$ a fixed positive integer?
After reading this question, I would like to just count all solutions for:
$$\frac{1}{2^{k_1}} + \frac{1}{2^{k_2}} + \frac{1}{2^{k_3}} + \dots + \frac{1}{2^{k_n}}=1$$
for $k_i\in \Bbb{N}$ (we can ...
- 2,103
6
votes
2
answers
4k
views
Find all ways to factor a number
An example of what I'm looking for will probably explain the question best. 24 can be written as:
12 · 2
6 · 2 · 2
3 · 2 · 2 · 2
8 · 3
4 · 2 · 3
6 · 4
I'm familiar with finding all the prime factors ...
- 169
6
votes
2
answers
241
views
$p\equiv 1\pmod 4\Rightarrow p=a^2+b^2$ and $p\equiv 1\pmod 8\Rightarrow p=a^2+2b^2$, what about for $p\equiv 1\pmod {2^n}$ in general
Primes $p$ with $p\equiv 1\pmod 4$ can be written as $p=a^2+b^2$ for some integers $a,b$. For $p\equiv 1\pmod 8$ we have $p=a^2+2b^2$. Can primes that satisfy $p\equiv 1\pmod{2^n}$ for $n>3$ be ...
- 1,950
4
votes
1
answer
188
views
Permutation Partition Counting
Consider the number $n!$ for some integer $n$
In how many ways can $n!$ be expressed as
$$a_1!a_2!\cdots a_n!$$
for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as $\...
- 17.5k
4
votes
3
answers
315
views
Computing Ramanujan asymptotic formula from Rademacher's formula for the partition function
I am trying to derive the Hardy-Ramanujan asymptotic formula
$$p(n) \sim \frac{1}{4n\sqrt{3}}e^{\pi\sqrt{\frac{2n}{3}}}$$
from Radmacher's formula for the partition function $p(n)$ given by
$$p(n)=\...
2
votes
1
answer
270
views
Related to the partition of $n$ where $p(n,2)$ denotes the number of partitions $\geq 2$
I was trying the following question from the Number theory book of Zuckerman.
If $p(n,2)$ denotes the number of partition of n with parts $\geq2$, prove that $p(n,2)>p(n-1,2)$ for all $n\geq 8$, ...
- 91
2
votes
1
answer
916
views
Number of solutions of Frobenius equation
I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$
where $x,y,z \in \{0,1,2,3,4,5,\dots\}$.
My attempt:
I noticed that that maximum value of $z$ could ...
- 22.5k
0
votes
0
answers
77
views
Factoring an integer $N$ using its random partition of length $3$
While working on this MSE question that I had posted, I wondered what would be a minimal base of numbers that we could work with the algorithm described in that question.
I conjectured that a ...
- 3,341