Mario Gionfriddo, Elena Guardo
The Erdös--Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x,y,z$ for every postive integers $n\geq 2$. In this short note we find explicity the solutions of the famous conjecture for the case $n\equiv13 \textrm{ mod } 24.$
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This paper makes a fundamental assertion about the Erdős-Straus conjecture. Suppose that for a prime $p$ there exists $x,y,z \in \mathbb{N}$ with $x \leq y \leq z$ so that $$ \frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}. $$ The main contribution of this paper is that, under this assumption, the Erdős-Straus conjecture can be reduced by one variable. For example, it is necessarily true that $$ z = \frac{xyp}{\gcd(y,p) \gcd \left( xy, x+y \right)}.$$ Considering other reductions of the Erdős-Straus conjecture, this paper suggests a method for proof.
This paper is a preliminary expository paper that outlines the relationship between solutions to the Erdős-Straus conjecture for a given prime $p$ and their corresponding Pythagorean triples. This paper also uses Bézout Coefficients to find a relationship between the solutions to the conjecture and roots of a second degree polynomial.
Further verification and empirical evidence for the Erdős-Straus conjecture
本文通过把计算验证上界提升至 10^18 并分析解计数函数 f(p)，为 Erdős-Straus 猜想提供新的经验支持。
Partial Resolution of the Erdös-Straus, Sierpinski, and Generalized Erdös-Straus Conjectures Using New Analytical Formulas
本文通过新的分析公式部分解决了埃德尔斯-斯特劳斯、西尔伯尼克及广义猜想，并探讨了其进一步证明的可能性。
A solution to the Straus-Erdős conjecture
本文提出了解决斯特劳斯-埃罗德猜想的方法，通过找到满足特定等式的正整数x、y、z来证明猜想成立。
Paul Erdos conjectured that for every n in N, n>1, there exist a, b, c natural numbers, not necessarily distinct, so that 4/n=1/a+1/b+1/c (see \cite{rg}). In this paper we prove an extension of Mordell's theorem and formulate a conjecture which is stronger than Erdos' conjecture.
The Erdös-Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x, y, z$ for every postive integers $n\ge 2$. We generalize the Erdös-Straus equation, state several methods for obtaining well-defined equations, and some features of well-defined equations. Using well-defined equations, we explain the inefficiency of the solutions that did not lead to a complete proof of the problem. Finally, we prove that this conjecture cannot be completely proven by the methods expressed in various articles. For this reason, we express con…
The aim of this note is to show that given a positive integer $n \geq 5$, the positive integral solutions of the diophantine equation $4/n = 1/x + 1/y+1/z$ cannot have solution such that $x$ and $y$ are coprime with $xy < \sqrt{z/2}$. The proof uses the continued fraction expansion of $4/n$.
In this paper we will explore the solutions to the diophantine equation in the Erdős-Straus conjecture. For a prime $p$ we are discussing the relationship between the values $x,y,z \in \mathbb{N}$ so that $$ \frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.$$ We will separate the types of solutions into two cases. In particular we will argue that the most common relationship found is $$ x = \lfloor \frac{py}{4y-p} \rfloor + 1.$$ Finally, we will make a few conjectures to motivate further research in this area.
In 1999 Allan Swett checked (in 150 hours) the Erdős-Straus conjecture up to $N=10^{14}$ with a sieve based on a single modular equation. After having proved the existence of a "complete" set of seven modular equations (including three new ones), this paper offers an optimized sieve based on these equations. A program written in C++ (and given elsewhere) allows then to make a checking whose running time, on a typical computer, range from few minutes for $N=10^{14}$ to about 16 hours for $N=10^{17}$.
We study the failure of the integral Hasse principle and strong approximation for the Erdős-Straus conjecture using the Brauer-Manin obstruction.
Elemental Patterns from the Erdős Straus Conjecture
本文基于Erdős Straus猜想提出一个新猜想，并证明新提出的模方程的解与原丢番图方程的解存在一一对应关系。
A Divisor Parametrization for the Erdős--Straus Conjecture
本文引入基于除数的函数 fab(n,a,b) 来参数化所有分母被 4 整除的 1/n 的三项埃及分数分解，并与已知的 Type I/II 描述进行比较。
An Unexpected Connection Between the Discrete Zeta Function and the Erdos-Straus Conjecture Under Mballa's Conjecture
本文在 Mballa 猜想下，将 $4(ζ(s)-1)$ 分解为由 Erdős–Straus 解构成的三项单位分数级数，从而建立离散 zeta 函数与埃及分数问题的联系。
A Complete Congruence System for the Erdos-Straus Conjecture
本研究围绕Erdos-Straus猜想展开，通过解的结构定义了涵盖所有素数的同余系统，提出A型、B型两类新解并关联对应同余式与多项式，拓展了该猜想的相关研究结果。
A unified parametric approach to the Erdős--Straus conjecture with explicit solutions for a set of integers of natural density one
本文提出参数方法研究丢番图方程，揭示其在埃尔多斯-斯特劳斯猜想中的结构行为，并为自然密度为一的整数集提供显式解。
Structure and form of the solutions of the Erdos-Straus conjecture
本文基于解的结构而非模相似性，对满足 Erdos-Straus 猜想的 p 值进行分类，刻画了 (du,dv,duv) 形式及 gcd(x,y,z)=x 的解，并处理了 p=1009 等难例。
For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with $x,y,z$ positive integers. The \emph{Erdős-Straus conjecture} asserts that $f(n) > 0$ for every $n \geq 2$. To solve this conjecture, it suffices without loss of generality to consider the case when $n$ is a prime $p$. In this paper we consider the question of bounding the sum $\sum_{p<N} f(p)$ asymptotically as $N \to \infty$, where $p$ ranges over primes. Our main result establishes the asymptotic upper and lower bounds $$ N \log^2 N…
A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples
提出取整连分数（FCT）框架构造Erdős-Straus猜想的三项埃及分数表示，利用移位整数p+i的除数结构而非同余技巧，推导出失败概率的超多项式上界，结合Borel-Cantelli引理启发式证明反例有限，并在10^17、10^52和10^131附近的10^9个素数上验证无反例。
In this paper we make a Gaussian integer version of the Erdős-Straus conjecture and we solve the Erdős-Straus diophantine equation over the rings of integers of norm-Euclidean quadratic fields.
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 …
Constructive Proofs of the Erdos-Straus Conjecture for Prime Numbers with P congruent to 1 modulo 4
本文针对 P ≡ 1 mod 4 的素数，提出两种构造性方法证明 Erdős–Straus 猜想，其中方法 ED2 将解集刻画为仿射格并显式构造表示，同时引入卷积与反卷积算法并通过大规模计算验证其正确性与有效性。
Almost a Complete Proof of the Generalized Erdős-Straus Conjecture: ${5}/{a} = {1}/{b} + {1}/{c} + {1}/{d}$
本文针对广义 Erdős-Straus 猜想，为所有整数 $a \ge 2$ 的 Diophantine 方程 $5/a=1/b+1/c+1/d$ 提供显式解，仅余 $q \equiv 0 \pmod{252}$ 部分提出经计算验证的多项式生成猜想。
Exact Polynomial Families Solving the Erdos-Straus Equation
针对 Erdős-Straus 猜想中未解的 $a \equiv 1 \pmod{4}$ 情形，本文构造了四个显式无界多变量多项式族，使前三个代入 $a = 4p_i+1$ 后恒有显式解，并通过高达 $10^9$ 的计算验证支持这四个多项式覆盖所有该同余整数的猜想。
Sharp bound for the Erdős-Straus non-averaging set problem
本文解决了 Erdős-Straus 问题，证明 $\{1,\ldots,n\}$ 的最大非平均子集大小为 $n^{1/4+o(1)}$，并推广到任意固定维度的盒子，主要工具是子集和结构定理与近似凸位置点集结构。
Exceptions to the Erd\H os--Straus--Schinzel conjecture
研究证明了对于足够大的m，存在某些n使得m/n无法表示为3个单位分数之和，并探讨了相关数值计算与依赖关系。
We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed $m$ there are at most $\mathcal{O}_ε(n^{3/5+ε})$ solutions of $\frac{m}{n}=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when $m=4$ and $n$ is a prime. Moreover there exists an algorithm finding all solutions in expected running time $\mathcal{O}_ε\left(n^ε\left(\frac{n^3}{m^2}\right)^{1/5}\right)$, for any $ε>0$. We also imp…
Potential Relation Between the Riemann Zeta Function and the Polynomial Function $F$ of the Generalized Erdős--Straus Conjecture, Subject to its Analytic Continuation
本文研究了Riemann泽塔函数与广义Erdős--Straus猜想多项式函数$F$之间的潜在关系，通过解析延拓建立了新的函数$G_k(s)$与泽塔函数的联系。
Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions
本文通过参数化同余类推导形如 24m+1 的整数的驯服解，覆盖了 m≤2000 时 591 个此类素数中的 586 个。
Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild Solutions
本文针对 Erdös-Straus 猜想，推导出三十四个野解族，覆盖了已知的十四个野素数，与驯化解结合可覆盖所有 $24m+1$ 形式的素数。
Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod{24}$
本文针对模24余1的素数，研究Erdős-Straus猜想的无因式分解搜索深度，证明了相关集合的上界，展示了无条件障碍，还提出了可判定区间内素数猜想的无因式分解方法，该方法存在I/II型遍历的代价不对称性。
Let $n$ be a positive integer. We study the diophantine equation $ab(ab-1)-na=Δ^2$, where $a,b$ are positive integers. We also show that if a system of two congruences is soluble, then an equation which is a translation of Erdős-Straus conjecture is soluble.
Quantitative correlations and some problems on prime factors of consecutive integers
本文研究了素因子数量函数ω(n)及相关函数Ω(n)和τ(n)的旧问题，证明了关于ω(n+k)的猜想，证明了ω(n)/2^n级数的无理性，并给出了ω(n)=ω(n+1)的渐进行为。
Decomposing a factorial into large factors
本文研究了将阶乘分解为N个大于等于t(N)整数的最大小数t(N)，并给出了更精确的渐进行为，同时验证了Guy和Selfridge的猜想。
Homogeneous structures in subset sums and non-averaging sets
本文证明大基数整数子集的子集和中必含高维齐次广义等差数列，并据此改进 Erdős–Straus 非平均集问题的阈值。
Bounded Representations by $x^2+y^2-z^2$
本文证明每个足够大的整数$n$均可表示为$n=x^2+y^2-z^2$，其中max(x²,y²,z²)≤n，利用二次型理论和Duke定理解决Erdős问题1148。
On the set of points represented by harmonic subseries
本文通过构造性方法解决 Erdős 和 Graham 提出的开放问题，证明由三类调和子级数构成的三维点集具有非空内部，推广了 Erdős 和 Straus 的二维结果。
Parametric Algorithms for the 5-Modular Analog of ES (Sierpiński): Structure of Solutions, Parameterization, and Constructive Proofs (SERP)
本文研究了将5/P表示为三个不同单位分数之和的问题，针对P≡1 mod5的情况，提出参数化构造算法并证明了解的密度特性。
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