Related papers: Approximating 1 from below using $n$ Egyptian frac…

We give a sharp upper bound for the entries of the representations of a rational number as a sum of Egyptian fractions.

An Egyptian fraction is a sum of the form $1/n_1 + \cdots + 1/n_r$ where $n_1, \dots, n_k$ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set $E_N$ of rational numbers that can be represented by…

Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.

A unit fraction representation of a rational number $r$ is a finite sum of reciprocals of positive integers that equals $r$. Of particular interest is the case when all denominators in the representation are distinct, resulting in an…

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N}…

This paper attempts to prove the Sylvester's conjecture using Egyptian Fractions with two key ingredients. First, creating a set of operators that completely generates all possible Egyptian fraction of 1. And second, to detect patterns in…

Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a…

Results about the structure of the set of Egyptian fractions on the line are extended to subsets of topological groups.

An Egyptian fraction is a sum of distinct unit fractions (reciprocals of positive integers). We show that every rational number has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest…

We consider the question of approximating any real number $\alpha$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on…

We explore a novel link between two seemingly disparate mathematical concepts: Egyptian fractions and fractals. By examining the decomposition of rationals into sums of distinct unit fractions, a practice rooted in ancient Egyptian…

Let $\mathcal{G}$ be the greedy algorithm that, for each $\theta\in (0,1]$, produces an infinite sequence of positive integers $(a_n)_{n=1}^\infty$ satisfying $\sum_{n=1}^\infty 1/a_n = \theta$. For natural numbers $p < q$, let…

We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.

Let $\mathcal A = (A_1,\ldots, A_n)$ be a sequence of nonempty finite sets of positive real numbers, and let $\mathcal{B} = (B_1,\ldots, B_n)$ be a sequence of infinite discrete sets of positive real numbers. A weighted real Egyptian number…

Given an integer partition of $n$ into distinct parts, the sum of the reciprocal parts is an example of an egyptian fraction. We study this statistic under the uniform measure on distinct parts partitions of $n$ and prove that, as $n \to…

It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…

This paper introduces a new equation for rewriting two unit fractions to another two unit fractions. This equation is useful for optimizing the elements of an Egyptian Fraction. Parity of the elements of the Egyptian Fractions are also…

Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a_1/q_1, ..., a_n/q_n with smaller denominators. We show that in the special cases of n=3 and n=4 and…

We present explicit formulas for the computation of the neighbors of several elements of Farey subsequences.

This paper provides bounds for the number of terms, denoted by $f$, of a harmonic sum with the condition that it starts from any arbitrary unit fraction $\frac{1}{m}$, $m > 1$, until another unit fraction $\frac{1}{m+f-1}$ such that the sum…