On Sierpiński and Riesel Repdigits and Repintegers
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| Format: | Preprint |
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2025
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| author | Bispels, Chris Cohen, Matthew Harrington, Joshua Lowrance, Joshua Pontes, Kaelyn Schaumann, Leif Wong, Tony W. H. |
| author_facet | Bispels, Chris Cohen, Matthew Harrington, Joshua Lowrance, Joshua Pontes, Kaelyn Schaumann, Leif Wong, Tony W. H. |
| contents | For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the case of $k=1$, we say that $1_b^{(t)}$ is a $b$-repunit. In this article, we investigate the existsence of $b$-repdigits and $b$-repunits among the sets of Sierpiński numbers and Riesel numbers. A Sierpiński number is defined as an odd integer $k$ for which $k\cdot 2^n+1$ is composite for all positive integers $n$ and Riesel numbers are similarly defined for the expression $k\cdot 2^n-1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Sierpiński and Riesel Repdigits and Repintegers Bispels, Chris Cohen, Matthew Harrington, Joshua Lowrance, Joshua Pontes, Kaelyn Schaumann, Leif Wong, Tony W. H. Number Theory 11A63, 11B25 For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the case of $k=1$, we say that $1_b^{(t)}$ is a $b$-repunit. In this article, we investigate the existsence of $b$-repdigits and $b$-repunits among the sets of Sierpiński numbers and Riesel numbers. A Sierpiński number is defined as an odd integer $k$ for which $k\cdot 2^n+1$ is composite for all positive integers $n$ and Riesel numbers are similarly defined for the expression $k\cdot 2^n-1$. |
| title | On Sierpiński and Riesel Repdigits and Repintegers |
| topic | Number Theory 11A63, 11B25 |
| url | https://arxiv.org/abs/2505.00778 |