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Questions tagged [egyptian-fractions]

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

6
votes
0answers
188 views

Product of sum of reciprocals

For any positive integers $k$ and $l$, does the equation $$(\sum_{i=1}^k \frac{1}{p_i}) (\sum_{j=1}^l \frac{1}{q_j}) = 1$$ have solutions in distinct primes, that is, $p_1, p_2, \dots, p_k, q_1, q_2, \...
4
votes
1answer
89 views

Egyptian Fraction when numerator is greater than denominator

I am doing an assignment about Egyptian fractions and I am a bit confused about what to do when the given fraction's numerator is greater than denominator. My initial idea was to subtract the fraction ...
1
vote
1answer
55 views

Permutations, products, and unit fractions

Here's a question motivated by some related MathOverflow questions of Zhi-Wei Sun. Show that, for any $n \ge 1$, there is a permutation of $\{1,2,\ldots, n\}$, i.e., some $\pi \in S_n$, such that $$\...
3
votes
1answer
39 views

Egyptian fractions: does the greedy algorithm never give more fractions than absolutely necessary?

Given $n > 1$, it's obvious that $$\frac{2^n - 1}{2^n} = \sum_{i = 1}^n \frac{1}{2^i}.$$ For example, $$\frac{15}{16} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}.$$ That's not the ...
0
votes
0answers
14 views

Partitioning a set of egyptian fractions into 2 groups which sum to at most 1.

Suppose we have a set of Egyptian fractions which sum to $X$. (Multiple of the same fraction is permitted). We try to partition them into two groups, such that each group sums to at most 1. What is ...
2
votes
1answer
140 views

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} = ...
0
votes
0answers
41 views

Counting Natural Solutions to Certain Quadratic Equations

I am interested in counting the number of distinct solutions (wlog, a < b) to this equation for a fixed value of y. $$\frac{x}{y} = \frac{1}{a} + \frac{1}{b}; a, b, y \in \mathbb{N}_+$$ I can ...
2
votes
0answers
72 views

Number of ways to sum two Egyptian fractions and satisfy a given inequality.

I am looking to finalize my solution to the following number theory problem: Find a rule in terms of n for the number of ways to write integers a and b, with a less than or equal to b, so that $$\...
12
votes
0answers
157 views

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 $a<b$, if we apply the greedy algorithm in order to express it as a sum of unit fractions, then we may get ...
5
votes
2answers
89 views

Is there a close form for $g(a,b,n)=\sum\limits_{k=0}^{n}\binom{n}{k}\frac{1}{ak+b}$?

We can be sure, that for $a>0$, $b>0$ $$f(a,b,n)=\sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{ak+b}=\frac{(an)!^{(a)}}{(an+b)!^{(a)}}$$ where $(an+b)!^{(a)}$ denotes multifactorial: $(n)!^{(1)}...
10
votes
1answer
663 views

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. ...
0
votes
2answers
58 views

Need assistance in finding sets of solutions for this equation

Given some integer $a$ and $n$, I want to find all integers $b$ and $t_i$ that satisfy this equation: $$ a = b - \frac{b}{t_1} - \frac{b}{t_2} - \cdots - \frac{b}{t_n} $$ I'm a little confused on ...
1
vote
0answers
60 views

$\frac{A}{B}$ as the sum of distinct inverse naturals.

Is there a method to taking some rational number and expressing it as the sum of distinct inverses of the natural numbers? For example, $$\frac{2}{3} = \frac{1}{6} + \frac{1}{2}$$ and $$\frac{47}{60} ...
3
votes
3answers
78 views

On the “hydra set” $S=\left\{\sum_{n\geq 1}\frac{1}{x_n}:(x_n)_{n\geq 1}\text{ is an increasing sequence of positive natural numbers}\right\}$

What are all the elements of set $$ S=\left\{ \sum_{n\geq 1}\frac{1}{x_n} \right\}$$ where $ (x_n)_{n\geq 1} $ is any increasing sequence of positive natural numbers for which $ \sum_{n\geq 1}\frac{1}...
3
votes
1answer
55 views

*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 ...
0
votes
1answer
51 views

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}...
1
vote
0answers
44 views

Explicit Bounds on the Lengths of Egyptian Fractions

Let $\frac{a}{b}$ be a rational number strictly between $0$ and $1$ , and let: $$\frac{a}{b}=\sum_{k=1}^{N}\frac{1}{n_{k}}$$ be an Egyptian fraction representation of $\frac{a}{b}$ . In my ...
5
votes
2answers
101 views

Egyptian fraction for $\varphi- {F(2n+2) \over F(2n+1)}$

The sum of the reciprocals of the ${2^n}$th Fibonacci numbers is known to be $\dfrac{3-\sqrt{5}}{2}$. https://math.stackexchange.com/a/746678/134791 This may be written as the following closed form ...
2
votes
1answer
95 views

Represent improper fraction as a sum of unique unit fractions

Is it possible to represent an improper fraction as a finite sum of unique unit fractions (Egyptian fractions)?
14
votes
1answer
398 views

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 ...
2
votes
1answer
64 views

Is this Egyptian fractions algorithm the same as the greedy algorithm? How to prove it?

I was thinking about possible new algorithms for Egyptian fractions expansion. We know that for any $p,q,m>0$: $$\frac{p}{q}>\frac{p}{q+m}$$ Here we assume $p<q$ and are coprime. Now it ...
4
votes
1answer
157 views

Why does Engel expansion give smaller denominators than the Greedy algorithm?

I compared the two best known algorithms for Egyptian fraction expansion: Greedy algorithm. On each step for a fraction $p_n/q_n$ we choose a denominator $a_n$ such that: $$\frac{p_n}{q_n}-\frac{1}{...
6
votes
1answer
295 views

Expanding integers into distinct egyptian fractions - what is the optimal way?

We know that there are infinite ways to represent $1$ as a sum of distinct unit fractions (i.e. egyptian fractions). The most optimal one (the least demoninators and the least number of fractions) is: ...
58
votes
0answers
867 views

$\frac{1}{n}$ as a difference of Egyptian fractions with all denominators $<n$

Is there a good characterization of the set $S$ of positive integers $n$ such that $\frac{1}{n}$ can be represented as a difference of Egyptian fractions with all denominators $< n$? For example, $...
1
vote
1answer
61 views

Egyptian fractions inequality problem

If $\frac{a}{b}$ is a non-unit fraction between 0 and 1, in lowest terms let $\frac{1}{n}$ be the largest unit fraction less than $\frac{a}{b}$, and form the fraction $\frac{a^{'}}{b^{'}}=\frac{a}{b}-\...
16
votes
1answer
248 views

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 ...
12
votes
0answers
143 views

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 ...
1
vote
2answers
112 views

Find all triples satisfying an equation [closed]

Another question I saw recently: Find all triples of positive integers $(a,b,c)$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Can someone help me with it?
6
votes
2answers
170 views

How many egyptian fractions including and above (1/n) are necessary to sum to unity

Let there be a finite set of positive integers such that: (a) no two members of the set are equal (b) the sum of the inverse of each member of the set is equal to one The smallest set (as defined ...
13
votes
0answers
256 views

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 ...
9
votes
1answer
158 views

Numbers whose reciprocals sum to $1$

What are all the numbers that can be written as $a_1+a_2+\dots+a_n$, where $a_1,\dots,a_n$ are positive integers such that $\frac{1}{a_1}+\dots+\frac{1}{a_n}=1$? For instance, such numbers include $4=...
3
votes
3answers
448 views

Find all integer solutions to $\frac{1}{x} + \frac{1}{y} = \frac{2}{3}$

Find all integer solutions $(x, y)$ of the equation $$\frac{1}{x} + \frac{1}{y} = \frac{2}{3}$$ What have done is that: $$\frac{1}{x}= \frac{2y-3}{3y}$$ so, $$x=\frac{3y}{2y-3}$$ If $2y-3 = +1 \...
4
votes
0answers
92 views

Egyptian fraction with least possible sum

Suppose that $~a~$ and $~b~$ are coprime positive integers. Then there exists representation of $~\frac{a}{b}~$ as egyptian fraction: $$~\frac{a}{b} = \frac{1}{d_1} + \cdots + \frac{1}{d_s} ~$$ There ...
3
votes
1answer
270 views

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 ...
1
vote
2answers
39 views

Under what conditions would the function $\prod_{i=1}^{n}{\frac{r_i}{r_i - 1}}$ be decreasing with respect to $n$?

So I know that $$\frac{r}{r - 1}$$ is a decreasing function of $r$. My question is: Under what conditions would the following function be decreasing with respect to $n$? $$\displaystyle \prod_{i=...
1
vote
2answers
48 views

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 $\...
3
votes
1answer
59 views

Prove that for all naturals $n \ge 6$ there is a set of $n$ positive naturals, $a_1$ to $a_n$ such that $\sum_{i=1}^n \left(\frac{1}{a_i}\right)^2 =1$

I don't know how to prove this. I know that $\{2, 2, 2, 2\}$ is a set for $n = 4$, since $\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)...
5
votes
4answers
610 views

Find All Dimensions such that Volume of Box = Surface Area

A rectangular prism has integer edge lengths. Find all dimensions such that its surface area equals its volume. My Attempt at a Solution: Let the edge lengths be represented by the variables $l, w, ...
64
votes
1answer
865 views

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 ...
3
votes
1answer
1k views

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 ...
1
vote
1answer
66 views

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 $n<p$.
4
votes
3answers
3k views

How do I prove that any unit fraction can be represented as the sum of two other distinct unit fractions?

A number of the form $\frac{1}{n}$, where $n$ is an integer greater than $1$, is called a unit fraction. Noting that $\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$ and $\frac{1}{3} = \frac{1}{4} + \frac{1}...
2
votes
1answer
118 views

A question on egyptian fractions

An Egyptian fraction is the sum of distinct unit fractions. Are there any 2000 egyptian fractions that their sum is 1?
3
votes
2answers
343 views

Egyptian fraction representations of real numbers

I've been looking into Egyptian fractions now, but information on certain topics seems scarce. Can you answer any of these questions that intrigue me: 1) What is known about the Egyptian fraction ...
1
vote
1answer
201 views

Any 'odd unit fraction' whose denominator is not $1$ can be represented as the sum of three different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is 1 'odd unit fraction'. Then, here is my question. Question : Is the following ...
10
votes
3answers
340 views

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.
4
votes
1answer
246 views

Algebraic structure of a set of Egyptian fractions of a positive rational?

It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions). I am struggling to understand in a formal way, ...
7
votes
3answers
2k views

Fractions in Ancient Egypt

In ancient Egypt, fractions were written as sums of fractions with numerator 1. For instance,$ \frac{3}{5}=\frac{1}{2}+\frac{1}{10}$. Consider the following algorithm for writing a fraction $\frac{m}{...
328
votes
18answers
36k views

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 ...
30
votes
3answers
2k views

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 ...