Runs of integers with constant values of the Carmichael function
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909219580542976 |
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| author | Lebowitz-Lockard, Noah Saunders, J. C. |
| author_facet | Lebowitz-Lockard, Noah Saunders, J. C. |
| contents | In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $λ(n+1)=λ(n+2)=\cdots=λ(n+k)$ holds for some $n\leq x$, where $λ$ is the Carmichael $λ$ function, is bounded above by $O\left((\log x\log\log x)^2\right)$. Their method involved bounding the value of $λ(n + i)$ from below using the prime factorization of $n + i$ for each $i \leq k$. They then used the fact that every $λ(n + i)$ had to satisfy this bound. Here we improve their result by incorporating a reverse counting argument on a result of Baker and Harman on the largest prime factor of a shifted prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Runs of integers with constant values of the Carmichael function Lebowitz-Lockard, Noah Saunders, J. C. Number Theory In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $λ(n+1)=λ(n+2)=\cdots=λ(n+k)$ holds for some $n\leq x$, where $λ$ is the Carmichael $λ$ function, is bounded above by $O\left((\log x\log\log x)^2\right)$. Their method involved bounding the value of $λ(n + i)$ from below using the prime factorization of $n + i$ for each $i \leq k$. They then used the fact that every $λ(n + i)$ had to satisfy this bound. Here we improve their result by incorporating a reverse counting argument on a result of Baker and Harman on the largest prime factor of a shifted prime. |
| title | Runs of integers with constant values of the Carmichael function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.06832 |