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Questions tagged [unit-fractions]

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21
votes
0answers
302 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,...
0
votes
1answer
154 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
1answer
265 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
1answer
155 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 ...
24
votes
1answer
738 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
1answer
400 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} $...
3
votes
0answers
117 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
1answer
786 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
0answers
437 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 ...
5
votes
4answers
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 $K_1,K_2\in$ {$1,\ldots,K$}, ...
17
votes
1answer
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 ...