Related papers: Denser Egyptian Fractions

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…

We answer several questions of P. Erdos and R. Graham by showing that for any rational number r > 0, there exist integers n1, n2, ..., nk, k variable, where N < n1 < n2 < ... < nk < (e^r + o_r(1) ) N, such that r = 1/n1 + 1/n2 + ... + 1/nk.…

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for…

If an irreducible fraction $\frac mn>0$ can be decomposed into the sum of several irreducible proper fractions with different denominators, and the positive number smaller than $\frac mn$ in fractional ideal $\frac 1n\mathbb Z$ can not be…

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erd\H{o}s-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than…

In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example,…

The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The…

We will discuss about a possible method of using the cubit rod by the architects and the surveyors of Ancient Egypt to measure and draw lengths, comparing it with the other interpretations present in Literature. Instead of the modern…

Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most $n/d$, adds the largest odd denominator unit fraction so the sum is at most $n/d$, and continues as…

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…

For a prime number $p$, let $A_3(p)= | \{ m \in \mathbb{N}: \exists m_1,m_2,m_3 \in \mathbb{N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} |$. In 2019 Luca and Pappalardi proved that $x (\log x)^3 \ll \sum_{p \le x} A_{3}(p)…

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…

Fracterms are introduced as a proxy for fractions. A precise definition of fracterms is formulated and on that basis reasonably precise definitions of various classes of fracterms are given. In the context of the meadow of rational numbers…

We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.

For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients…

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions…

We show that each rational number $r$, $0\le r<1$, occurs as the fractional part of a Dedekind sum $S(m,n)$.

Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant…

We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…