Carmichael Numbers in All Possible Arithmetic Progressions
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917014410362880 |
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| author | Larsen, Daniel |
| author_facet | Larsen, Daniel |
| contents | We prove that every arithmetic progression either contains infinitely many Carmichael numbers or none at all. Furthermore, there is a simple criterion for determining which category a given arithmetic progression falls into. In particular, if $m$ is any integer such that $(m,2ϕ(m))=1$ then there exist infinitely many Carmichael numbers divisible by $m$. As a consequence, we are able to prove that $\liminf_{n\text{ Carmichael}}\frac{ϕ(n)}{n}=0$, resolving a question of Alford, Granville, and Pomerance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Carmichael Numbers in All Possible Arithmetic Progressions Larsen, Daniel Number Theory 11N25 We prove that every arithmetic progression either contains infinitely many Carmichael numbers or none at all. Furthermore, there is a simple criterion for determining which category a given arithmetic progression falls into. In particular, if $m$ is any integer such that $(m,2ϕ(m))=1$ then there exist infinitely many Carmichael numbers divisible by $m$. As a consequence, we are able to prove that $\liminf_{n\text{ Carmichael}}\frac{ϕ(n)}{n}=0$, resolving a question of Alford, Granville, and Pomerance. |
| title | Carmichael Numbers in All Possible Arithmetic Progressions |
| topic | Number Theory 11N25 |
| url | https://arxiv.org/abs/2504.09056 |