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Questions tagged [egyptian-fractions]

Writing positive rational numbers as the sum of fractions with all numerators equal to one.

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Is the reciprocal golden ratio well approximated by this exponentially sparse series of reciprocal Fibonacci numbers?

Let $1/\phi= \phi-1\approx0.618\,$ denote the reciprocal golden ratio and $\mathrm F(k)\;(k=0,1,...)$ the Fibonacci numbers, where $\mathrm F(0)=0,\mathrm F(1)=1,$ and $\mathrm F(k+1)=\mathrm F(k)+\...
John Bentin's user avatar
1 vote
2 answers
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Do Egyptian representations follow a matroid structure?

Let $V=(v_1,\cdots,v_n), v_i \in \mathbb N$ be independent with respect to $\frac p q$ if there doesn't exist an indicator vector $\beta = (b_1, \cdots, b_n), b_i \in \{0,1\}$ with $\sum \frac{b_i}{...
Snared's user avatar
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4 votes
4 answers
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Expressing $\frac{2}{n}$ as the sum of two unit fractions

Consider fractions such as $\frac{2}{5}$ and $\frac{2}{7}$ expressed as the sum of two unit fractions. Respectively, they can be expressed as $\frac{1}{3}+\frac{1}{15}$ and $\frac{1}{4}+\frac{1}{28}$. ...
James Chadwick's user avatar
9 votes
1 answer
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A game of magic Egyptian tilings

Background I've recently been formulating a game that incorporates elements from Egyptian fractions, magic squares, and tilings. It is a single-player game in which the objective is to tessellate a ...
Max Muller's user avatar
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2 votes
1 answer
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$m_i, n_j$ integers and $\{m_i\}_{i=1}^{k}\neq\{n_j\}_{j=1}^{k'}.$ Does $\sum\frac{1}{m_i}=\sum\frac{1}{n_j}\implies\sum m_i\neq\sum n_j?$

Suppose $\{m_i\}_{i=1}^{k}$ and $\{n_j\}_{j=1}^{k'}$ are each finite subsets of $\mathbb{N},$ $\{m_i\}_{i=1}^{k}\neq\{n_j\}_{j=1}^{k'},$ and $\displaystyle\sum_{i=1}^{i=k}\frac{1}{m_i} = \sum_{j=1}^{...
Adam Rubinson's user avatar
10 votes
2 answers
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Conjecture: If $\sum_{i=1}^n \frac{1}{x_i}=1$ then $x_i | x_j$

Let $x_1,x_2,x_3,\cdots , x_n\in\mathbb{N}$ prove that if $\sum_{i=1}^n\frac{1}{x_i}=1$, then there exist $x_j,x_i$ such that $x_j | x_i$. I read that I should avoid the no clue questions, but this is ...
Moaoly's user avatar
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4 votes
2 answers
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Has the diophantine equation $ \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right) $ been studied?

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 curious about solutions ...
Max Muller's user avatar
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2 votes
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Egyptian fraction of a number in the interval (0.5,1)

Assume the real number $a$ such that $0.5 < a < 1$ and $a$ can be expressed as an Egyptian fraction of length $l$, wich means for natural numbers $n_1$ to $n_l$ we have: $$ a = \sum^l_1{\frac{1}{...
Peyman's user avatar
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Egyptian fraction of length 3 in the interval (0.5,1)

I've been exploring the properties of Egyptian fractions, which are representations of numbers as sums of distinct unit fractions. Specifically, I'm interested in numbers $a$ such that $ 0.5 < a &...
Peyman's user avatar
  • 770
4 votes
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Have the prime Egyptian fractionary expansions of rational numbers been studied before?

Background 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}=...
Max Muller's user avatar
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Expressing one as a sum of unique unit fractions whose denominators are increasing?

Given lower bound $N\in\mathbb N$, I am wondering if it is possible to express $1$ as a finite sum of unique unit fractions: $$1=\frac{1}{n_1}+\frac{1}{n_2}+\cdots+\frac{1}{n_k}$$ where $$N\le n_1<...
John Davies's user avatar
2 votes
1 answer
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Four different positive integers whose reciprocals sum to 1

https://oeis.org/A006585 says The 6 solutions for n=4 are 2,3,7,42; 2,3,8,24; 2,3,9,18; 2,3,10,15; 2,4,5,20; 2,4,6,12. How would one prove 42 is indeed the largest here? And/or this list is ...
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What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?

Motivation In the following question posted both here on MSE and over at MO, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
Max Muller's user avatar
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5 votes
3 answers
482 views

Is there a finite set of distinct naturals whose reciprocals sum to 1, and no element in the set is one less than a prime?

I'll try to state this formally, forgive me if I botch the notation. Let $$R=\{n : n+1 \in\mathbb{P}\}.$$ (Where $\mathbb{P}$ is the set of primes) Then the question is, $$\exists S \subset (\mathbb{N}...
Trevor's user avatar
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2 votes
1 answer
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Reference Request - Online copy of R. Graham's "On Finite Sums of Rational Numbers"

Does anyone know where I could find a copy of Ron Graham's PhD dissertation, titled "On Finite Sums of Rational Numbers"? I found this paper, with a similar title, but I could not find ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
0 answers
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$A = \sum_{i=1}^{\infty} \frac{i}{a_i}$ and $1 = \sum_{i=1}^{\infty} \frac{1}{a_i}$?

I have the following problem : For a given positive real $A>1$ $$A = \sum_{i=1}^{\infty} \frac{i}{a_i}$$ $$1 = \sum_{i=1}^{\infty} \frac{1}{a_i}$$ $$a_{n+1}>a_{n}$$ where $a_i$ is a strictly ...
mick's user avatar
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3 votes
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The longest sequence of unique unit fractions that sums to 1, given a specific bound on the denominators

Suppose one is tasked to find the longest sequence of unique unit fractions (1/x) that sums to 1, given that 2 <= x <= 100. Currently, I am going with a decomposition approach. For example, by ...
Div4Expert's user avatar
1 vote
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how to find number of presentation of 1 as a sum of $2^{-i}$? [duplicate]

How to find number of presentations of 1 as a sum of exactly $k$ numbers of the form $2^{-i}$ ? As an example for $k=2$ we have only one presentation: $$1 = \frac{1}{2} + \frac{1}{2},$$ so answer for $...
rrryok's user avatar
  • 51
9 votes
1 answer
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Shortest palindromic Egyptian representation for reciprocal integers

Consider the problem of representing the reciprocal of an integer as an Egyptian fraction where all the denominators are palindromes. i.e. write $$ \frac{1}{n} = \sum_{i} \frac{1}{a_i} $$ where $a_i$...
Peder's user avatar
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Let $n\ge2$ be an integer and $S=${$1,\cdots,n$}. For which $\ T\subset S\ $ is $\sum_{m\in T} \frac{1}{m}$ less than $1$, but as large as possible?

Let $n\ge 2$ be an integer and $S=${$1,\cdots,n$}. For which subset $T$ of $S$ is the sum $$\sum_{m\in T} \frac{1}{m}$$ smaller than $1$ , but as large as possible ? Example : $n=30$. Brute force ...
Peter's user avatar
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2 votes
0 answers
57 views

Can we always use denominator $2$ for an optimal solution?

Let $r$ be a rational number with $\frac{1}{2}<r<1$ Let $(a_1,\cdots,a_k)$ be a solution of $r=\frac{1}{a_1}+\cdots +\frac{1}{a_k}$ with distinct positive integers $a_1,\cdots , a_k$ , in other ...
Peter's user avatar
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the smallest number n that so that $\exists $ distinct $a_1,\cdots, a_n \in (2\mathbb{Z}_{\ge 1} + 1), \sum_{i=1}^n \dfrac{1}{a_i}=1$

This is PUMAC $2008$ Division A Number Theory problem $10$. What is the smallest number n that so that $\exists $ distinct $a_1,\cdots, a_n \in (2\mathbb{Z}_{\ge 1} + 1), \sum_{i=1}^n \dfrac{1}{a_i}=...
user33096's user avatar
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1 vote
1 answer
207 views

Show that $\exp(f(z)) = \frac{1}{1−z}$

I'm self teaching Complex Analysis, and am working on this question: Show that the series expansion of $\exp(z)$ has radius of convergence 1. Let $f(z)$ be the function to which the series $\sum\...
yw_2003's user avatar
  • 319
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0 answers
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Why the Rhind papyrus contains representations only to fractions of the form $2/n$?

The Rhind papyrus contained a table with representations as sums of unit fractions for each of the fractions $2/n$, where $n$ is an odd number from $3$ to $101$. For example, $$ \frac{2}{5} = \frac{1}{...
sdd's user avatar
  • 451
2 votes
0 answers
110 views

Mathematical applications of Egyptian fractions

Background In chapter 5 of the book "Number Theory in Science and Communication" by Manfred E. Schroeder, the author goes into continued, Egyptian and Farey fractions. On p. 65, he writes: &...
Max Muller's user avatar
  • 7,148
6 votes
1 answer
166 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
  • 7,148
3 votes
0 answers
87 views

Rational Engel or Egyptian zeta expansions: what is known?

The Engel expansion of a positive real number $x$ is the unique non-decreasing sequence of positive integers $\{a_{1},a_{2},a_{3},\dots \} $ such that $$x=\frac{1}{a_{1}} + \frac{1}{a_{1} a_{2}} + \...
Max Muller's user avatar
  • 7,148
5 votes
0 answers
128 views

A weighted count of Egyptian fraction representations

Given a positive rational $\alpha$ and a natural number $k$, let $N_k(\alpha)$ be the number of Egyptian fraction representations of $\alpha$ with smallest element ${1\over k}$ (see e.g. the survey &...
Noah Schweber's user avatar
8 votes
0 answers
243 views

Weighted count of Egyptian fraction representations

This question emerged during an activity I ran for some middle school students a couple weeks ago; basically, it's about a way to "count" - with an appropriate kind of weight - the Egyptian ...
Noah Schweber's user avatar
4 votes
1 answer
190 views

Egyptian unit decomposition + coverings of $\mathbb{Z}_P$ by arithmetic progressions

For a quick exposition of how I arrived at this question: I found the Wikipedia page on covering sets and through some experiments on my own (fixing the constant at the end, the factor $k$ itself etc.)...
TheOutZ's user avatar
  • 1,266
8 votes
0 answers
195 views

Forming rational numbers using unique Egyptian fractions, all but one of whom have coprime denominators

Question: For a given rational number $r\in (0,1)$, does there exists a finite $S\subset \mathbb{N}$ such that every pair of elements of $S$ are coprime and $$r-\sum_{n\in S}\frac{1}{n}=\frac{1}{b}$$ ...
QC_QAOA's user avatar
  • 11.9k
10 votes
2 answers
155 views

If $\sum_n \frac{1}{a_n} = 2$ where $a_n$ are positive integers, is there a subset such that $\sum_{n\in S} \frac{1}{a_n} = 1$? [duplicate]

As the title says: I'm wondering, out of curiosity, whether any (weak) Egyptian fraction decomposition of 2 always splits into two Egyptian fraction decompositions of 1. By a "weak" ...
Daniel Schepler's user avatar
3 votes
0 answers
200 views

Is there a way to prove if an Egyptian fraction is optimal?

Egyptian fractions are the representation of a rational number (for example, $\frac{61}{66}$) as a sum of reciprocals of integers (with that example, $2^{-1} + 3^{-1} + 11^{-1}$), a.k.a. unit ...
Lawton's user avatar
  • 1,861
1 vote
2 answers
125 views

Finding perfect two terms Egyptian fraction

If I have two unity fractions, like $\frac{1}{12} + \frac{1}{180}$, for instance. These two fractions can be re-writen as $\frac{1}{15} + \frac{1}{45}$ or even $\frac{1}{18} + \frac{1}{30}$, which ...
Adam's user avatar
  • 87
1 vote
0 answers
175 views

Can Greedy algorithm (egyptian fractions) never halt?

The Greedy Algorithm seems a standard way of computing egyptian fractions, but I can't find any proof that it always halts nor I can prove it. Is there any algorithm for egyptian fractions that ...
Carlos Eduardo's user avatar
0 votes
1 answer
61 views

Is this condition sufficient for an egyptian fraction to be greedy

An egyptian fraction $\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} ...$ is greedy if each $a_n$ is as small as possible, given its predecessors. Every real number between 0 and 1 has exactly one ...
dspyz's user avatar
  • 870
1 vote
1 answer
88 views

Is this egyptian fractions problem NP-complete?

Given a set of positive integers $\ M=${$\ a_1,a_2,\cdots ,a_k\ $} and a rational number $\ r\ $ , is the following decision problem NP-complete ? Is there a subset $S\subset M$ with $$\sum_{p\in S} \...
Peter's user avatar
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4 votes
0 answers
71 views

Given a representation with egyptian fractions, how to check efficiently whether it is optimal?

The fraction $\ \frac{a}{b}\ $ with coprime integers $\ a,b\ $ with $\ 1<a<b\ $ is supposed to be represented as a sum of pairwise distinct egyptian fractions, for example $$\frac{12}{13}=\frac{...
Peter's user avatar
  • 85.1k
4 votes
0 answers
170 views

What is known about this extension of the Erdös-Strauss-conjecture?

The Erdös-Strauss conjecture states that $$\frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ has a solution in positive integers for every integer $n>1$. What is known about this extension : The ...
Peter's user avatar
  • 85.1k
3 votes
2 answers
319 views

Find the number of all triplets (x, y, z) of positive integers such that 1/x + 1/y + 1/z = 1/2021.

1.Find the number of all triplets $(x, y, z)$ of positive integers such that $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2021}.$$ 2.How about if $x,y,z$ are odd? What I did: we can get $$xyz=2021(...
Mr.He's user avatar
  • 579
0 votes
1 answer
144 views

how to measure the work done to calculate the Egyptian (2/n) fractions of the Rhind papyrus through an algorithm?

How the Egyptian fractions (2/n) collected by the scribe Ahmes that are contained in the Rhind papyrus were obtained? The work to obtain them must have been as hard as carrying stones to build a ...
Wilson Massaro's user avatar
9 votes
1 answer
408 views

How can I represent a fraction as a finite sum of reciprocal squares?

I've found this result : $$ \frac{1}2 = \frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\frac1{5^2}+\frac1{6^2}+\frac1{15^2}+\frac1{18^2}+\frac1{36^2}+\frac1{60^2}+\frac1{180^2}$$ I've tried my best to make a ...
Jean Pierre's user avatar
0 votes
3 answers
421 views

Odd perfect numbers and egyptian fraction conjecture

I have reached the conclusion that the non-existence of odd perfect numbers is related to the following Conjecture Let it be $R=\{d_1,d_2,...,d_n\}$ the set of distinct proper divisors less than $\...
Juan Moreno's user avatar
  • 1,190
1 vote
2 answers
325 views

Proving a necessary condition for an egyptian fraction with odd denominators

In the question posted in Proving an equality involving cyclic sums, I realized that all the possible solutions to the following egyptian fraction with denominators from a set of odd positive ...
Juan Moreno's user avatar
  • 1,190
1 vote
1 answer
45 views

Set of distinct odd positive integers such that $\left(\frac{1}{O_{1}}+\frac{1}{O_{2}}+\frac{1}{O_{3}}+...+\frac{m+1}{O_{n}}\right)=1$

I am looking for a set of distinct odd positive integers $3\leq{O_1}<O_2<O_3<...<O_n$ such that $$\frac{1}{O_{1}}+\frac{1}{O_{2}}+\frac{1}{O_{3}}+...+\frac{m+1}{O_{n}}=1$$ such that $m\in\...
Juan Moreno's user avatar
  • 1,190
0 votes
2 answers
100 views

Find minimum $n$ that satisfies $\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=\frac{12}{13}$

From the test: We have the following equation: \begin{equation} \frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=\frac{12}{13} \end{equation} where $a_i$ are distinct natural numbers not equal to $13$....
Lee's user avatar
  • 1,910
6 votes
1 answer
91 views

$\frac{1}{d_1} + \dots + \frac{1}{d_k} = 1,$ and $\gcd(d_i,d_j)>1 \, \forall i,j$ implies $\gcd(d_1, \dots, d_k) > 1$ for distinct $d_i.$

Conjecture: If $d_i \in \mathbb{N}$ are distinct, $\frac{1}{d_1} + \dots + \frac{1}{d_k} = 1,$ and $\gcd(d_i,d_j)>1 \, \forall i,j,$ then $\gcd(d_1, \dots, d_k) > 1.$ Motive: In the process of ...
Display name's user avatar
  • 5,230
4 votes
3 answers
229 views

Positive integer solutions to $\frac{1}{a} + \frac{1}{b} = \frac{c}{d}$

I was looking at the equation $$\frac{1}{a}+\frac{1}{b} = \frac{c}{d}\,,$$ where $c$ and $d$ are positive integers such that $\gcd(c,d) = 1$. I was trying to find positive integer solutions to this ...
mihirb's user avatar
  • 802
3 votes
4 answers
325 views

How do I find the integer solutions that satisfy $xyz = 288$ and $xy + xz + yz = 144$?

Find all integers $x$, $y$, and $z$ such that $$xyz = 288$$ and $$xy + xz + yz = 144\,.$$ I did this using brute force, where $$288 = 12 \times 24 = 12 \times 6 \times 4$$ and found that these set of ...
aco's user avatar
  • 515
1 vote
0 answers
36 views

Optimization on the Decomposition of 1 to unit fractions with $\frac{1}{5}$ as the largest part.

This is a follow question to the link: On the decomposition of $1$ as the sum of Egyptian fractions with odd denominators - Part II Suppose we relax the condition that any term can be divisible by 3 ...
Keneth Adrian Dagal's user avatar