A Complete Congruence System for the Erdos-Straus Conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913316428840960 |
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| author | Lopez, Miguel Angel |
| author_facet | Lopez, Miguel Angel |
| contents | In this paper we attack the Erdos-Straus conjecture by means of the structure of its solutions, extending and improving the results of a previous paper. Using previous results and supported by the works of Elsholtz and Tao and Monks and Velingker we define a system of congruences for which there are always solutions to the Erdos-Straus conjecture and which we conjecture to include all prime numbers. For this purpose, and always taking into account a result due to Mordell that limits the congruences admitting polynomial identities to those that are not quadratic residues, we will adopt a transversal approach and classify the solutions by their form and not by those congruences that produce them. Thus we define two new types of solutions, which we call Type A and B, and relate them to the already known Type II solutions and study their properties. Finally we conjecture that every prime number has at least one solution of Type A or B and we associate a congruence and a general polynomial to each Type of solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_01508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Complete Congruence System for the Erdos-Straus Conjecture Lopez, Miguel Angel Number Theory 11D72, 11A07, 11A41 In this paper we attack the Erdos-Straus conjecture by means of the structure of its solutions, extending and improving the results of a previous paper. Using previous results and supported by the works of Elsholtz and Tao and Monks and Velingker we define a system of congruences for which there are always solutions to the Erdos-Straus conjecture and which we conjecture to include all prime numbers. For this purpose, and always taking into account a result due to Mordell that limits the congruences admitting polynomial identities to those that are not quadratic residues, we will adopt a transversal approach and classify the solutions by their form and not by those congruences that produce them. Thus we define two new types of solutions, which we call Type A and B, and relate them to the already known Type II solutions and study their properties. Finally we conjecture that every prime number has at least one solution of Type A or B and we associate a congruence and a general polynomial to each Type of solution. |
| title | A Complete Congruence System for the Erdos-Straus Conjecture |
| topic | Number Theory 11D72, 11A07, 11A41 |
| url | https://arxiv.org/abs/2404.01508 |