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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.06057 |
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| _version_ | 1866908695375380480 |
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| author | Seraj, Samer |
| author_facet | Seraj, Samer |
| contents | We observe that the computation $5^2 = 25$ has the digital property of the result being equal to the exponent concatenated directly to the left of the base. The generalization to a Diophantine equation and inequality in number bases has been articulated previously, but a comprehensive answer was not available in the literature. We classify and largely parametrize the solutions. Tools that play key roles are the Newton-Raphson method, the arithmetic-geometric means inequality, Pell's equation, and Fermat's little theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polymorphic Numbers with Exponential Prefix Seraj, Samer History and Overview 11A63, 11D09, 11A07, 26A06 We observe that the computation $5^2 = 25$ has the digital property of the result being equal to the exponent concatenated directly to the left of the base. The generalization to a Diophantine equation and inequality in number bases has been articulated previously, but a comprehensive answer was not available in the literature. We classify and largely parametrize the solutions. Tools that play key roles are the Newton-Raphson method, the arithmetic-geometric means inequality, Pell's equation, and Fermat's little theorem. |
| title | Polymorphic Numbers with Exponential Prefix |
| topic | History and Overview 11A63, 11D09, 11A07, 26A06 |
| url | https://arxiv.org/abs/2512.06057 |