Almost a Complete Proof of the Generalized Erdős-Straus Conjecture: ${5}/{a} = {1}/{b} + {1}/{c} + {1}/{d}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913982982389760 |
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| author | Ghermoul, Bilal |
| author_facet | Ghermoul, Bilal |
| contents | The generalized Erdős-Straus conjecture, proposed by Wacław Sierpiński in 1956, asks whether the Diophantine equation \[ \frac{5}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b,c,d \in \mathbb{N}$ for every integer $a \ge 2$. In this work we present explicit solutions for all integers $a \ge 2$. We begin with the simplest known cases where $a \equiv i \pmod{5}$ for $i \in \{0,2,3,4\}$, providing direct decompositions. The remaining open case, $a \equiv 1 \pmod{5}$, is addressed for $a = 5q + 1$ with $q \not\equiv 0 \pmod{252}$, where we give explicit decompositions, often with $q$ expressed as three-variable polynomials. For $q \equiv 0 \pmod{252}$, we conjecture that a specific polynomial $p_{1}(x,y,z)=z (x (5 y-1)-y)-x,~ x,y,z \in \mathbb{N}^*$, which exactly satisfies the generalized Erdős--Straus equation, generates all such multiples of $252$. This conjecture has been verified computationally for $5q+1$ up to approximately $10^{10}$, and the corresponding \textit{Mathematica} implementation is included. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_07367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost a Complete Proof of the Generalized Erdős-Straus Conjecture: ${5}/{a} = {1}/{b} + {1}/{c} + {1}/{d}$ Ghermoul, Bilal Number Theory The generalized Erdős-Straus conjecture, proposed by Wacław Sierpiński in 1956, asks whether the Diophantine equation \[ \frac{5}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b,c,d \in \mathbb{N}$ for every integer $a \ge 2$. In this work we present explicit solutions for all integers $a \ge 2$. We begin with the simplest known cases where $a \equiv i \pmod{5}$ for $i \in \{0,2,3,4\}$, providing direct decompositions. The remaining open case, $a \equiv 1 \pmod{5}$, is addressed for $a = 5q + 1$ with $q \not\equiv 0 \pmod{252}$, where we give explicit decompositions, often with $q$ expressed as three-variable polynomials. For $q \equiv 0 \pmod{252}$, we conjecture that a specific polynomial $p_{1}(x,y,z)=z (x (5 y-1)-y)-x,~ x,y,z \in \mathbb{N}^*$, which exactly satisfies the generalized Erdős--Straus equation, generates all such multiples of $252$. This conjecture has been verified computationally for $5q+1$ up to approximately $10^{10}$, and the corresponding \textit{Mathematica} implementation is included. |
| title | Almost a Complete Proof of the Generalized Erdős-Straus Conjecture: ${5}/{a} = {1}/{b} + {1}/{c} + {1}/{d}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.07367 |