Questions tagged [unit-fractions]
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20 questions
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A puzzle with magic Egyptian tilings
Background
I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...
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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 ...
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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 ...
- 7,226
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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 $$\frac{...
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Egyptian fractions similar to Erdos-Straus conjecture
It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions.
My question is whether it is known that if $a>4$
$$
\frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}
$...
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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
$$r=\sum_{p\in P_r^-}\frac1{p-1}=\sum_{p\in ...
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Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?
It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since
$$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,...
- 7,003
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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 ...
- 21.1k
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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:
$$\...
- 8,842
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A conjecture on covers of $\mathbb Z$ by residue classes
Let $A=\{a_s+n_s\mathbb Z\}_{s=1}^k$ be a finite system of residue classes, where $a_s$ and $n_s>0$ are integers. For a positive integer $m$, if $A$ covers each integer at least $m$ times then we ...
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Maximum in solution set to a Diophantine equation related to unit fractions
Some time ago, Kellogg communicated to Carmichael a result with an incomplete proof, which was soon after verified as correct. I do not recall the source but recall the result. Define
$$S_n = \{ (x_1,...
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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 ...
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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 ...
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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 $\...
CommunityBot
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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 ...
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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, \...
- 44.4k
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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 \...
CommunityBot
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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 ...
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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 ...
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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 $K_1,K_2\in$ {$1,\ldots,K$}, ...
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