Articles of egyptian fractions

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 $\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$, find a general result of the form $\frac{1}{n} = \frac{1}{a} + \frac{1}{b}$ and hence prove that any unit fraction can be expressed as […]

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: $$\frac{p_n}{q_n}-\frac{1}{a_n} \geq 0, \quad \frac{p_n}{q_n}-\frac{1}{a_n-1} < 0$$ $$\frac{p_{n+1}}{q_{n+1}}=\frac{a_n p_n-q_n}{q_n a_n}$$ Engel expansion. On each step for a fraction $p_n/q_n$ we choose a multiplier $m_n$ such that: $$\frac{m_n p_n}{q_n} […]

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 $\frac{m}{n}$ in this form$(1\leq m < n)$: write the fraction $\frac{1}{\lceil n/m\rceil}$ , calculate the fraction $\frac{m}{n}-\frac{1}{\lceil n/m \rceil}$ , and if it is nonzero repeat the same step. Prove that […]

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 having any specific algorithm, I got as far as the picture below, which serves more to get a feel for what I […]

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 The author proves a very important theorem, which I will present here with changes to suit […]

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 defined: $$x_n-\frac{1}{a_n}>0$$ $$x_n-\frac{1}{a_n-1}<0$$ And $x_n$ are defined: $$x_{n+1}=x_n-\frac{1}{a_n}$$ This expansion may rival the simple continued fractions in its importance to the number theory. It’s unique for every number and […]

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 representation (by the greedy algorithm) of irrational numbers? Are Egyptian fractions known to be interesting in any similar sense as the continued fractions? […]

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 problem to a 5th grade child. (I’m a college student tutor) This girl goes to a […]