Carmichael numbers and least common multiples of $p-1$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916455671398400 |
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| author | Wright, Thomas |
| author_facet | Wright, Thomas |
| contents | For a Carmichael number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-1,\cdots,p_m-1],$$ and let $C_ν(X)$ denote the number of Carmichael numbers up to $X$ such that $K=ν$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any even natural number $ν$, $$C_ν(X)\geq X^{1-(2+o(1))\frac{\log\log\log \log X}{\log\log\log X}}.$$ This is a departure from standard constructions of Carmichael numbers, which generally require $K$ to grow along with $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_16397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Carmichael numbers and least common multiples of $p-1$ Wright, Thomas Number Theory 11A51 For a Carmichael number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-1,\cdots,p_m-1],$$ and let $C_ν(X)$ denote the number of Carmichael numbers up to $X$ such that $K=ν$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any even natural number $ν$, $$C_ν(X)\geq X^{1-(2+o(1))\frac{\log\log\log \log X}{\log\log\log X}}.$$ This is a departure from standard constructions of Carmichael numbers, which generally require $K$ to grow along with $n$. |
| title | Carmichael numbers and least common multiples of $p-1$ |
| topic | Number Theory 11A51 |
| url | https://arxiv.org/abs/2409.16397 |