Brauer-Manin obstruction for Erdős-Straus surfaces

A short proof of the conjecture of Erdös--Straus for every $n\equiv 13 \textrm{ mod }24$

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 $n\equiv13 \textrm{ mod } 24.$

A remark on a conjecture of Erdős and Straus

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 $4/n$.

A Geometric Consideration of the Erdős-Straus Conjecture

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.

The Erdős-Straus conjecture New modular equations and checking up to $N=10^{17}$

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 $N=10^{17}$.

来自Erdős Straus猜想的元素模式

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。

Erdős–Straus 猜想的一个除数参数化

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。

关于埃尔多斯-斯特劳斯猜想的统一参数方法及自然密度为一的整数集的显式解

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。

Erdős-Straus猜想的取整连分数方法：反例的启发式有限性

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。

Unit Fractions in Norm-Euclidean Rings of Integers

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.

Erdős-Straus 非平均集问题的尖锐界

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。

On Egyptian fractions

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 …

广义 Erdős-Straus 猜想的几乎完整证明：${5}/{a} = {1}/{b} + {1}/{c} + {1}/{d}$

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。

求解 Erdos-Straus 方程的精确多项式族

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。

埃尔多斯-斯特劳斯-辛内尔猜想的例外情况

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。

The number of solutions of the Erdős-Straus Equation and sums of $k$ unit fractions

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…

Riemann泽塔函数与广义Erdős--Straus猜想多项式函数$F$之间潜在关系的研究，基于其解析延拓

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。

支持 Erdös-Straus 猜想的同余类 I: 驯服解

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。

支持 Erdös-Straus 猜想 的同余类 II：野解

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。

模24余1的Erdős-Straus猜想的筛维与搜索深度

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。