# Questions tagged [unit-fractions]

The unit-fractions tag has no usage guidance.

14
questions

**1**

vote

**1**answer

277 views

### A conjecture on covers of $\mathbb Z$ by residue classes

Let $A=\{a_s+n_s\mathbb Z\}_{s=1}^k$ be a finite system of residue classes, where $a_s$ and $n_s>0$ are integers. For a positive integer $m$, if $A$ covers each integer at least $m$ times then we ...

**0**

votes

**1**answer

70 views

### Maximum in solution set to a Diophantine equation related to unit fractions

Some time ago, Kellogg communicated to Carmichael a result with an incomplete proof, which was soon after verified as correct. I do not recall the source but recall the result. Define
$$S_n = \{ (x_1,...

**26**

votes

**0**answers

466 views

### Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?

It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since
$$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,...

**1**

vote

**1**answer

219 views

### Is it true that $\sum_{k=m}^n\frac{\sigma(k)}k\not\in\mathbb Z$ for all derangements $\sigma\in S_n$ and $1\le m\le n$?

Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$.
Recall that a permutation $\sigma\in S_n$ is called a derangemnt if $\sigma(k)\not=k$ for all $k=1,\ldots,n$.
Motivated ...

**-1**

votes

**1**answer

289 views

### Odd & even permutations and unit fractions

One more motivated by recent questions of Zhi-Wei Sun.
Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$.
Is it true that, for every $n \ge 8$, there is at least one even permutation $\...

**1**

vote

**1**answer

172 views

### Derangements and unit fractions

Motivated by a recent question of Zhi-Wei Sun and its nice answer by Zhao Shen, here are two related questions.
Let $S_n$ be the group of permutations on $\{1, 2, \ldots, n\}$.
a. For each $n \ge ...

**25**

votes

**1**answer

788 views

### Only finitely many values of the symmetric functions of $1/1,1/2,\ldots,1/n$ are $2$-adic integers (?)

For integers $n \geq k \geq 1$ let
$$H(n,k) := \sum_{1 \leq i_1 < \cdots < i_k \leq n} \frac1{i_1 \cdots i_k}$$
be the $k$-th elementary symmetric function of $\tfrac1{1},\tfrac1{2}, \ldots, \...

**4**

votes

**1**answer

467 views

### Egyptian fractions similar to Erdos-Straus conjecture

It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions.
My question is whether it is known that if $a>4$
$$
\frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}
$...

**4**

votes

**0**answers

127 views

### On the comparison of Egyptian fractions of two kinds

I posted the question on MSE here but it did not get any answer.
Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset \...

**1**

vote

**1**answer

915 views

### On unitary fractions

My apologies if the question has already been discussed somewhere else, I did not found anything related to unitary fractions with the search tool...
It is a nice exercise for high-school students to ...

**3**

votes

**0**answers

466 views

### State of ignorance concerning Erdos-Straus

The Erdos-Straus Conjecture says that, for all $n > 1$, there exist positive integers $x,y,z$, such that $\dfrac{4}{n} = \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$. A generalization due to ...

**6**

votes

**4**answers

1k views

### The difference of two sums of unit fractions

I had this question bothering me for a while, but I can't come up with a meaningful answer.
The problem is the following:
Let integers $a_i,b_j\in${$1,\ldots,n$} and $K_1,K_2\in$ {$1,\ldots,K$}, ...

**13**

votes

**3**answers

852 views

### Unit fraction, equally spaced denominators not integer

I've been looking at unit fractions, and found a paper by Erdős "Some properties of partial sums of the harmonic series" that proves a few things, and gives a reference for the following theorem:
$$\...

**20**

votes

**1**answer

1k views

### What's the simplest rational not expressible as a sum of a given number of unit fractions?

This is essentially the same as the closed question Representation of rational numbers as the sum of 1/k but I hope I can make a case for it as an MO-worthy question.
Ed Pegg, Jr., in his Math Games ...