On binomial coefficients associated with Sierpiński and Riesel numbers
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arXiv
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| Auteurs principaux: | , , , , , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866916803705307136 |
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| author | Armbruster, Ashley Barger, Grace Bykova, Sofya Dvorachek, Tyler Eckard, Emily Harrington, Joshua Sun, Yewen Wong, Tony W. H. |
| author_facet | Armbruster, Ashley Barger, Grace Bykova, Sofya Dvorachek, Tyler Eckard, Emily Harrington, Joshua Sun, Yewen Wong, Tony W. H. |
| contents | In this paper, we investigate the existence of Sierpiński numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpiński numbers and Riesel numbers of the form $\binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1\leq r\leq x$ for which $\binom{k}{r}$ is a Sierpiński number for infinitely many $k$. We further show that the value $S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations to base $a$-Sierpiński numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $\binom{k}{r}$ is simultaneously a base $a$-Sierpiński and base $a$-Riesel number for infinitely many $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_08085 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On binomial coefficients associated with Sierpiński and Riesel numbers Armbruster, Ashley Barger, Grace Bykova, Sofya Dvorachek, Tyler Eckard, Emily Harrington, Joshua Sun, Yewen Wong, Tony W. H. Number Theory 11A07, 11B65 In this paper, we investigate the existence of Sierpiński numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpiński numbers and Riesel numbers of the form $\binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1\leq r\leq x$ for which $\binom{k}{r}$ is a Sierpiński number for infinitely many $k$. We further show that the value $S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations to base $a$-Sierpiński numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $\binom{k}{r}$ is simultaneously a base $a$-Sierpiński and base $a$-Riesel number for infinitely many $k$. |
| title | On binomial coefficients associated with Sierpiński and Riesel numbers |
| topic | Number Theory 11A07, 11B65 |
| url | https://arxiv.org/abs/2010.08085 |