# Questions tagged [egyptian-fractions]

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

116 questions
Filter by
Sorted by
Tagged with
46 views

• 972
171 views

### 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 $\frac{1}{4}+\frac{1}{28}$. ...
145 views

### 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 ...
• 7,148
79 views

• 51
145 views

### 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 $a_i$...
• 2,192
1 vote
51 views

### 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 ...
• 85.1k
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 ...
• 85.1k
68 views

• 319
86 views

• 7,148
87 views

• 85.1k
71 views

• 579
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 ...
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 ...
421 views

• 1,190
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: $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=\frac{12}{13}$$ where $a_i$ are distinct natural numbers not equal to $13$....
• 1,910
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 ...
• 5,230
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 ...
• 802
### 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 ...
### 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 ...