Generalized Sierpiński and Riesel numbers of the form $tb^t+α$
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| Format: | Preprint |
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2025
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| _version_ | 1866911098095009792 |
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| author | Evans, Hailey Harrington, Joshua Heiney, Kendall Wieczorek, Maggie |
| author_facet | Evans, Hailey Harrington, Joshua Heiney, Kendall Wieczorek, Maggie |
| contents | Let $b\geq 2$ be an integer. We call an integer $k$ a $b$-Sierpiński number if $\gcd(k+1,b-1)=1$ and $k\cdot b^n+1$ is composite for all positive integers $n$. We similarly call $k$ a $b$-Riesel number if $\gcd(k-1,b-1)=1$ and $k\cdot b^n-1$ is composite for all positive integers $n$. An integer that is simultaneously $b$-Sierpiński and $b$-Riesel is called a $b$-Brier number. In this article, we show that for any integer $α\neq 0$, there are infinitely many $b$-Sierpiński numbers and infinitely many $b$-Riesel numbers of the form $tb^t+α$. We further show that when $b+1$ is not a power of $2$, there are infinitely $b$-Brier number of this form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Sierpiński and Riesel numbers of the form $tb^t+α$ Evans, Hailey Harrington, Joshua Heiney, Kendall Wieczorek, Maggie Number Theory 11A07 Let $b\geq 2$ be an integer. We call an integer $k$ a $b$-Sierpiński number if $\gcd(k+1,b-1)=1$ and $k\cdot b^n+1$ is composite for all positive integers $n$. We similarly call $k$ a $b$-Riesel number if $\gcd(k-1,b-1)=1$ and $k\cdot b^n-1$ is composite for all positive integers $n$. An integer that is simultaneously $b$-Sierpiński and $b$-Riesel is called a $b$-Brier number. In this article, we show that for any integer $α\neq 0$, there are infinitely many $b$-Sierpiński numbers and infinitely many $b$-Riesel numbers of the form $tb^t+α$. We further show that when $b+1$ is not a power of $2$, there are infinitely $b$-Brier number of this form. |
| title | Generalized Sierpiński and Riesel numbers of the form $tb^t+α$ |
| topic | Number Theory 11A07 |
| url | https://arxiv.org/abs/2508.05942 |