Unit fractions with shifted prime denominators
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911372778930176 |
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| author | Bloom, Thomas F. |
| author_facet | Bloom, Thomas F. |
| contents | We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm{ prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash\{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02689 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unit fractions with shifted prime denominators Bloom, Thomas F. Number Theory We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm{ prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash\{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied. |
| title | Unit fractions with shifted prime denominators |
| topic | Number Theory |
| url | https://arxiv.org/abs/2305.02689 |