On the decimal and octal digits of $1/p$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915960194072576 |
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| author | Girstmair, Kurt |
| author_facet | Girstmair, Kurt |
| contents | Let $p$ be a prime $\equiv 3$ mod 4, $p>3$, and suppose that 10 has the order $(p-1)/2$ mod p. Then $1/p$ has a decimal period of length $(p-1)/2$. We express the frequency of each digit $0,\ldots,9$ in this period in terms of the class numbers of two imaginary quadratic number fields. We also exhibit certain analogues of this result, so for the case that 10 is a primitive root mod $p$ and for the octal digits of $1/p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the decimal and octal digits of $1/p$ Girstmair, Kurt Number Theory Let $p$ be a prime $\equiv 3$ mod 4, $p>3$, and suppose that 10 has the order $(p-1)/2$ mod p. Then $1/p$ has a decimal period of length $(p-1)/2$. We express the frequency of each digit $0,\ldots,9$ in this period in terms of the class numbers of two imaginary quadratic number fields. We also exhibit certain analogues of this result, so for the case that 10 is a primitive root mod $p$ and for the octal digits of $1/p$. |
| title | On the decimal and octal digits of $1/p$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.07873 |