Questions tagged [unit-fractions]

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The diophantine equation $ \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right) $

Background I wonder if there are any rational numbers such that their Egyptian fraction (sum) representations are equal to their Egyptian product analogue. In other words, I am curious1 about ...
Max Muller's user avatar
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13 votes
1 answer
391 views

Egyptian fraction of a number in the interval (0.5,1)

An Egyptian fraction is a finite sum of distinct unit fractions, such as $$\frac{43}{48} = \frac{1}{2}+\frac{1}{3}+\frac{1}{16}.$$ Does there exist a number in the range $(0.5, 1)$ that when written ...
Peyman's user avatar
  • 243
14 votes
1 answer
392 views

Product analogue of Egyptian fractions

Background An Egyptian fraction is a finite sum of distinct unit fractions, in which each denominator is not bigger than the next one. In other words, it is a representation of $a/b$ such that $$\frac{...
Max Muller's user avatar
  • 4,435
2 votes
0 answers
85 views

Ratio of the number of solutions to unit fraction equations with shifted prime and natural denominators

In a 2018 question posed by Zhi-Wei Sun, he conjectures that for any rational number $r>0$, there are finite sets $P_r^-$ and $P_r^+$ of primes such that $$r=\sum_{p\in P_r^-}\frac1{p-1}=\sum_{p\in ...
Max Muller's user avatar
  • 4,435
10 votes
0 answers
244 views

A weighted count of Egyptian fraction representations

Previously asked and bountied at MSE: Given a positive rational $q$, let $$\mathsf{E}(q)=\left\{X\in[\mathbb{N}]^{\mathit{fin}}: q=\sum_{x\in X}{1\over x}\right\}$$ be the set of Egyptian fraction ...
Noah Schweber's user avatar
1 vote
1 answer
290 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 ...
Zhi-Wei Sun's user avatar
  • 14.4k
0 votes
1 answer
106 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,...
Descartes Before the Horse's user avatar
32 votes
1 answer
762 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,...
Zhi-Wei Sun's user avatar
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1 vote
1 answer
232 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 ...
Zhi-Wei Sun's user avatar
  • 14.4k
-1 votes
1 answer
375 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 $\...
Brian Hopkins's user avatar
1 vote
1 answer
181 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 ...
Brian Hopkins's user avatar
25 votes
1 answer
825 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, \...
user avatar
5 votes
2 answers
571 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} $...
asad's user avatar
  • 841
5 votes
0 answers
135 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 \...
Shivam Patel's user avatar
2 votes
1 answer
951 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 ...
Nekochan's user avatar
  • 449
3 votes
0 answers
519 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 ...
Woett's user avatar
  • 1,494
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 $K_1,K_2\in$ {$1,\ldots,K$}, ...
Anadim's user avatar
  • 449
15 votes
3 answers
974 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: $$\...
mmm's user avatar
  • 305
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 ...
Gerry Myerson's user avatar