Elemental Patterns from the Erdős Straus Conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911810866642944 |
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| author | Bradford, Kyle |
| author_facet | Bradford, Kyle |
| contents | This paper makes the following conjecture: For every prime $p$ there exists a positive integer $x$ with $\left\lceil \frac{p}{4} \right\rceil \leq x \leq \left\lceil \frac{p}{2} \right\rceil$ and a positive divisor $d|x^2$ so that either: (1) $ d \bmod \left( 4x - p \right) \equiv -px$; or (2) $d \leq x$ and $ d \bmod \left( 4x - p \right) \equiv -x$. Furthermore this paper proves that the solutions to these modular equations are in one-to-one correspondence with the solutions of the diophantine equation used in the Erdős Straus conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elemental Patterns from the Erdős Straus Conjecture Bradford, Kyle Number Theory 11A99 This paper makes the following conjecture: For every prime $p$ there exists a positive integer $x$ with $\left\lceil \frac{p}{4} \right\rceil \leq x \leq \left\lceil \frac{p}{2} \right\rceil$ and a positive divisor $d|x^2$ so that either: (1) $ d \bmod \left( 4x - p \right) \equiv -px$; or (2) $d \leq x$ and $ d \bmod \left( 4x - p \right) \equiv -x$. Furthermore this paper proves that the solutions to these modular equations are in one-to-one correspondence with the solutions of the diophantine equation used in the Erdős Straus conjecture. |
| title | Elemental Patterns from the Erdős Straus Conjecture |
| topic | Number Theory 11A99 |
| url | https://arxiv.org/abs/2403.16047 |