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 denominator, improving a result from an earlier paper to best-possible form. We also settle, in best-possible form, the related problem of how small M_t(r) can be such that there is an Egyptian fraction representation of r with exactly t terms, the denominators of which are all at most M_t(r). We also consider the following problems posed by Erdos and Graham: the set of integers that cannot be the largest denominator of an Egyptian fraction representation of 1 is infinite - what is its order of growth? How about those integers that cannot be the second-largest (third-largest, etc.) denominator of such a representation? In the latter case, we show that only finitely many integers cannot be the second-largest (third-largest, etc.) denominator of such a representation; while in the former case, we show that the set of integers that cannot be the largest denominator of such a representation has density zero, and establish its order of growth. Both results extend to representations of any positive rational number.
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.
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 very sparse subset of the positive integers up to the largest demoninator. We show that for every positive rational there exist Egyptian fractions whose largest denominator is at most N and whose denominators form a positive proportion of the integers up to N, for sufficiently large N; furthermore, the proportion is within a small factor of best possible.
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 $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.
In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.
From varying Egyptian fraction equations we obtain generalizations of primary pseudoperfect numbers and Giuga numbers which we call prime power psuedoperfect numbers and prime power Giuga numbers respectively. We show that a sequence of Amarnath Murthy in the OEIS is a subsequence of the sequence of prime power psuedoperfect numbers. Prime factorization conditions sufficient to imply a number is a prime power pseudoperfect number or a prime power Giuga number are given. The conditions on prime factorizations naturally give rise to a generalization of Fermat primes which we call extended Fermat…
Resolving a conjecture of Zhi-Wei Sun, we prove that every rational number can be represented as a sum of distinct unit fractions whose denominators are practical numbers. The same method applies to allowed denominators that are closed under multiplication by two and include a multiple of every positive integer, including the odious numbers, evil numbers, Hardy-Ramanujan numbers, Jordan-Polya numbers, and fibbinary numbers.
An increasing sequence $(x_i)_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation of $θ\in (0,1]$ if $\sum_{i=1}^n \frac{1}{x_i} < θ$. A greedy algorithm constructs an $n$-term underapproximation of $θ$. For some but not all numbers $θ$, the greedy algorithm gives a unique best $n$-term underapproximation for all $n$. An infinite set of rational numbers is constructed for which the greedy underapproximations are best, and numbers for which the greedy algorithm is not best are also studied.
Given a positive integer $n$ we let $A_k(n)$ be the number of positive integers $a$ such that $\frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}$ for some $m_1,m_2,\ldots,m_k\in {\mathbb N}$. We show that $x(\log x)^3\ll \sum_{p\le x} A_3(p)\ll x(\log x)^5$ as $x\rightarrow\infty$.
Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers $m_1, \dotsc, m_k$ is bounded above by $n^{1 - 1/2^{k-2} + o(1)}$ as $n$ goes to infinity. The proof is elementary.
The number of solutions of the diophantine equation $\sum_{i=1}^k \frac{1}{x_i}=1,$ in particular when the $x_i$ are distinct odd positive integers is investigated. The number of solutions $S(k)$ in this case is, for odd $k$: \[\exp \left( \exp \left( c_1\, \frac{k}{\log k}\right)\right) \leq S(k) \leq \exp \left( \exp \left(c_2\, k \right)\right) \] with some positive constants $c_1$ and $c_2$. This improves upon an earlier lower bound of $S(k) \geq \exp \left( (1+o(1))\frac{\log 2}{2} k^2\right)$.
For fixed integer $a\ge3$, we study the binary Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular the number $E_a(N)$ of $n\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}}$$ as $N$ goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of the underlying group structure.
In this article, we continue with our investigation of the Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions $R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\log 3\log\log n$ and variance $(log 3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects.
For given positive integers $n$ and $a$, let $R(n;\,a)$ denote the number of positive integer solutions $(x,\,y)$ of the Diophantine equation $$ {a\over n}={1\over x}+{1\over y}. $$ Write $$ S(N;\,a)=\sum_{\substack{n\leq N (n,\,a)=1}}R(n;\,a). $$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\leq N$ and $a\leq 2N$, there is an asymptotic formula $$ S(N;\,a)={3\over π^2a}\prod_{p|a}{p-1\over p+1}\cdot N(\log^2N+c_1(a) \log N+c_0(a))+Δ(N;\,a). $$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$.
We find a polynomial in three variables whose values at nonnegative integers satisfy the Erdős-Straus Conjecture. Although the perfect squares are not covered by these values, it allows us to prove that there are arbitrarily long sequence of consecutive numbers satisfying the Erdős-Straus Conjecture. We conjecture that the values of this polynomial include all the prime numbers of the form $4q+5$, which is checked up to $10^{14}$. A greedy-type algorithm to find an Erdős-Straus decomposition is also given; the convergence of this algorithm is proved for a wide class of numbers. Combining this …
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 with numerators $\mathcal{A}$ and denominators $\mathcal{B}$ is a real number $c$ that can be represented in the form \[ c = \sum_{i=1}^n \frac{a_i}{b_i} \] with $a_i \in A_i$ and $b_i \in B_i$ for $i \in \{1,\ldots, n\}$. In this paper, classical results of Sierpinski for Egyptian fractions are extended to the set of weighted real Egyptian numbers.
The Erdös-Moser equation is a Diophantine equation proposed more than 60 years ago which remains unresolved to this day. In this paper, we consider the problem in terms of divisibility of power sums and in terms of certain Egyptian fraction equations. As a consequence, we show that solutions must satisfy strong divisibility properties and a restrictive Egyptian fraction equation. Our studies lead us to results on the Bernoulli numbers and allow us to motivate Moser's original approach to the problem.
Let $T_o(k)$ denote the number of solutions of $\sum_{i=1}^k\frac 1{x_i}=1$ in odd numbers $1<x_1<x_2<...<x_k$. It is clear that $T_o(2k)=0$. For distinct primes $p_1, p_2,..., p_t$, let $S(p_1, p_2,..., p_t)=\{p_1^{α_1}...p_t^{α_t}\mid α_i\in \mathbb{N}_0, i=1,2,..., t}$. Let $T_k(p_1,..., p_t)$ be the number of solutions $\sum_{i=1}^{k}\frac 1{x_i}=1$ with $1<x_1<x_2<...<x_{k}$ and $x_i\in S(p_1, p_2,..., p_t)$. It is clear that if $T_k(p_1,..., p_t)\not= 0$ for some $k$, then the inverse sum of all elements $s_j>1$ in $S(p_1, p_2,..., p_t)$ is more than 1. In this paper we study $T_o(k)$ an…
