Constructive Proofs of the Erdos-Straus Conjecture for Prime Numbers with P congruent to 1 modulo 4
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911258504069120 |
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| author | Dyachenko, E. |
| author_facet | Dyachenko, E. |
| contents | The Erdos-Straus conjecture (ESC) concerns the representation of the fraction 4/P, where P is a prime number, as a sum of three positive unit fractions. The focus here is on the case when P is congruent to 1 modulo 4. Two constructive approaches are proposed. Method ED1 is based on a factorization identity and leads to a nonlinear parameterization in P, which requires divisor enumeration and local filtering. Method ED2 yields a linear system in P for the parameters (delta, b, c), describing the solution set as an affine lattice of finite index in Z^3. The central result states that for every prime P congruent to 1 modulo 4 there exists a representation: 4/P = 1/A + 1/(bP) + 1/(cP), where the triple (delta, b, c) in N^3 is constructed explicitly by method ED2. In addition, algorithms for transforming solutions (convolution and anti-convolution) are introduced, and large-scale computational verification confirms the correctness and efficiency of the proposed methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07465 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructive Proofs of the Erdos-Straus Conjecture for Prime Numbers with P congruent to 1 modulo 4 Dyachenko, E. Number Theory 11N05, 11P21, 94A60, 20P05 The Erdos-Straus conjecture (ESC) concerns the representation of the fraction 4/P, where P is a prime number, as a sum of three positive unit fractions. The focus here is on the case when P is congruent to 1 modulo 4. Two constructive approaches are proposed. Method ED1 is based on a factorization identity and leads to a nonlinear parameterization in P, which requires divisor enumeration and local filtering. Method ED2 yields a linear system in P for the parameters (delta, b, c), describing the solution set as an affine lattice of finite index in Z^3. The central result states that for every prime P congruent to 1 modulo 4 there exists a representation: 4/P = 1/A + 1/(bP) + 1/(cP), where the triple (delta, b, c) in N^3 is constructed explicitly by method ED2. In addition, algorithms for transforming solutions (convolution and anti-convolution) are introduced, and large-scale computational verification confirms the correctness and efficiency of the proposed methods. |
| title | Constructive Proofs of the Erdos-Straus Conjecture for Prime Numbers with P congruent to 1 modulo 4 |
| topic | Number Theory 11N05, 11P21, 94A60, 20P05 |
| url | https://arxiv.org/abs/2511.07465 |