标题： 精确多项式族解欧拉-斯特劳斯方程 显示英文标题

标题： Exact Polynomial Families Solving the Erdos-Straus Equation

摘要： 埃拉托色尼-斯特劳斯猜想，由保罗·埃尔德什和恩斯特·G. 斯特劳斯于1948年提出，询问对于每个整数$a \geq 2$，不定方程\[ \frac{4}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \]是否存在正整数解$b, c, d \in \mathbb{N}^*$。虽然该猜想已被证实对所有偶数和所有与$3 \pmod{4}$同余的整数成立，但情况$a \equiv 1 \pmod{4}$仍然是主要的开放挑战。 在这项工作中，我们构造了四个显式的无界多变量多项式$p_1(x,y,z), p_2(x,y,z), p_3(x,y,z), p_4(x,y,z)$与$x, y, z \geq 1$，使得前三个中的每一个——当插入形式$a = 4p_i(x,y,z)+1$——总是产生$a$的值，对于这些值，Erdős--Straus 方程存在显式解。因此，前三个多项式各自对其所有输出满足该猜想。我们进一步猜想，值\[ 4p_1(x,y,z)+1,\quad 4p_2(x,y,z)+1,\quad 4p_3(x,y,z)+1,\quad 4p_4(x,y,z)+1 \]共同覆盖所有与$1 \pmod{4}$同余的整数。 大量计算验证至$q = 10^9$确认，在此范围内的所有形式为$4q+1$的整数都至少来自这些族之一。 其中一个多项式单独生成所有这样的素数值，至少到$1.2 \times 10^{10}$。 这些结果为该猜想的解决提供了有力的计算证据和显式构造。 显示英文摘要

摘要： The Erd\H{o}s-Straus conjecture, proposed in 1948 by Paul Erd\H{o}s and Ernst G. Straus, asks whether the Diophantine equation \[ \frac{4}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b, c, d \in \mathbb{N}^*$ for every integer $a \geq 2$. While the conjecture has been confirmed for all even integers and for all integers congruent to $3 \pmod{4}$, the case $a \equiv 1 \pmod{4}$ remains the central open challenge. In this work, we construct four explicit unbounded multivariable polynomials $p_1(x,y,z), p_2(x,y,z), p_3(x,y,z), p_4(x,y,z)$ with $x, y, z \geq 1$, such that each of the first three -- when inserted into the form $a = 4p_i(x,y,z)+1$ -- always produces values of $a$ for which the Erd\H{o}s--Straus equation admits an explicit solution. Thus, the first three polynomials individually satisfy the conjecture for all their outputs. We further conjecture that the values \[ 4p_1(x,y,z)+1,\quad 4p_2(x,y,z)+1,\quad 4p_3(x,y,z)+1,\quad 4p_4(x,y,z)+1 \] collectively cover all integers congruent to $1 \pmod{4}$. Extensive computational verification up to $q = 10^9$ confirms that every integer of the form $4q+1$ within this range arises from at least one of these families. One of the polynomials alone generates all such prime values up to at least $1.2 \times 10^{10}$. These results offer strong computational evidence and explicit constructions relevant to the resolution of the conjecture.