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Questions tagged [egyptian-fractions]

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

Writing positive rational numbers as the sum of fractions with all numerators equal to one, i.e. $\frac1n$. The denominators are usually required to be distinct.

So, for example, $1=\frac12+\frac13+\frac16$.

See Wikipedia:Egyptian Fraction and Mathworld:Egyptian Fraction for more information.

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*Disjoint* Egyptian Fraction representations of $1$

I was doing a bit of reading about Egyptian Fractions. For those not familiar with the concept, an Egyptian Fraction is a sum of distinct unit fractions, or reciprocals of positive integers. The text that I read argued that since the number $1$ has…
Franklin Pezzuti Dyer
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Sets of Egyptian fractions which sum to 1

Let an 'Egyptian unity sum set' be a set of positive integers {a, b, c ...} such that their Egyptian fractions sum to 1; and none of the elements are equal. That is: 1/a + 1/b + 1/c ... = 1 Let the number of elements in any such set be equal to…
Thomas Delaney
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Real world applications of the study of Egyptian Fractions

I recently read about the study of Egyptian Fractions on the Good Math, Bad Math blog. The references to this article show that many years of research have gone into trying to find efficient ways to calculate the minimum length forms. Are there any…
Jen S.
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Representing positive integers with Egyptian fractions as short as possible

Supposing n a positive integer (1,2,3,...), we call Eg(n) the smallest number of different reciprocals (1,$\frac{1}{2}$,$\frac{1}{3}$,...) that sum up to n. We call also $\delta(n)$ the number of ways we can write n with Eg(n) reciprocals. We can…
Rami Rami
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Show Sylvester sequence is the smallest solution with n terms to sum of unit fractions equalling 1

I want to show that a prefix of Sylvester's sequence gives the "smallest" solution to the equation where the sum of n unit fractions equals 1. $$\sum_{i=1}^{n-1}{\frac{1}{x_i}} + \frac{1}{x_n - 1} = 1$$ Where the terms can be defined in two…
spyr03
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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 exhaustive?
chx
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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 satisfies the Egyptian Fraction concept of…
Adam
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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}
Juan Moreno
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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 with the largest term of the decomposition as…
Keneth Adrian Dagal
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If $0 < \frac{a}{b} < 1$, does subtracting the next largest number $\frac{1}{n}$ always make the resulting fraction's numerator less than $a$?

Assuming a domain of the natural numbers, if $\frac{a}{b}$ is a non-unit fraction between $0$ and $1$ in lowest terms, and $\frac{1}{n}$ is the largest unit fraction less than $\frac{a}{b}$, and $\frac{a'}{b'} = \frac{a}{b} - \frac{1}{n}$, then is…
Dylan Nissley
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Prove a sufficient condition for $\frac{n}{p}$ to have an egyptian fraction expansion of length $2$.

Prove that $\frac{n}{p}$ has an egyptian fraction expansion of length $2$ if and only if $n|(p+1)$ where $p$ is a odd prime and $n
Satvik Mashkaria
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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 representation as a greedy egyptian…
dspyz
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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 pyramid. For the same fraction, sometimes, we have…
Wilson Massaro
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Sums of Egyptian Fractions of Minimal Length

Let $\frac{a}{b}$ and $\frac{p}{q}$ be rational numbers in the interval $\left(0,1\right)$ such that $\frac{a}{b}+\frac{p}{q}<1$, and such that: $$\frac{a}{b} = \frac{1}{u_{1}}+\cdots+\frac{1}{u_{M}}$$ $$\frac{p}{q} =…
MCS
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