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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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Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was surprised that a teacher would assign this kind of…
Low Scores
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Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a known regular way to do this? Just playing and not…
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Can the sums of two sequences of reciprocals of consecutive integers be equal?

I'm primarily a programmer, so forgive me if I don't know the proper nomenclature or notation. Last night, an old teacher of mine told me about a question that had caused some noodle-scratching for him: For any two sequences of consecutive integers,…
Justin Myles Holmes
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Numbers $p-\sqrt{q}$ having regular egyptian fraction expansions?

I remind that the greedy algorithm for egyptian fraction expansion for a positive number $x_0 <1$ goes like this: $$x_0=\frac{1}{a_0}+\frac{1}{a_1}+\frac{1}{a_2}+\dots$$ $a_n$ are positive integers and are…
Yuriy S
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Surprising continued fractions of numbers in the form $\sum_{n=0}^\infty \frac{1}{a^{2^n}}$, including the same pattern for every $a>2$

I've been interested in the numbers of this form because it can be proved that for integer $a \geq 2$ all of them are irrational: $$x_a=\sum_{n=0}^\infty \frac{1}{a^{2^n}}$$ They satisfy the conditions listed in this paper: The Approximation of…
Yuriy S
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Egyptian fractions with very large denominators

It is well-known that if we have a fraction $\frac ab$, with $a,b\in\mathbb N$ and $a
José Carlos Santos
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Why this algorithm for egyptian fractions doesn't terminate in ~$2$% cases?

I thought up yet another algorithm for egyptian fraction expansion which turned out to be very effective (in terms of the length and the denominator size) - in most cases. However, for some fractions it doesn't terminate at all - it leads to an…
Yuriy S
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Greedy algorithm Egyptian fractions for irrational numbers - patterns and irrationality proofs

This is related to another question on this site, but it's not a duplicate, because the actual questions I ask are completely different. In one of the answers Jeffrey Shallit provided a very useful link on the topic…
Yuriy S
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Math contest proof problem fractions

Could someone help me with this? Let $x, y, z$ be positive integers with greatest common divisor $1$. If $\frac 1 x +\frac 1 y=\frac 1 z$, then show that $\sqrt{x + y}$ is an integer.
user87611
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Sufficiently large integers can be partitioned into squares of distinct integers whose reciprocals sum to 1.

OEIS sequence A297895 describes Numbers that can be partitioned into squares of distinct integers whose reciprocals sum to 1. 1, 49, 200, 338, 418, 445, 486, 489, 530, 569, 609, 610, 653, 770, 775, 804, 845, 855, 898, 899, 939, 978, 1005, 1019,…
Peter Kagey
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If $\sum_n \frac{1}{a_n} = 2$ where $a_n$ are positive integers, is there a subset such that $\sum_{n\in S} \frac{1}{a_n} = 1$?

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" Egyptian fraction decomposition, I mean one where the denominators…
Daniel Schepler
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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 $a_i$ is a palindrome (repeating $a_i$ is allowed). …
Peder
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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 fraction representations of a given rational. Let…
Noah Schweber
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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 $$r-\sum_{n\in S}\frac{1}{n}=\frac{1}{b}$$ for some $b\in\mathbb{N}$? Example: For example,…
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