Fibonacci primes, primes of the form $2^n-k$ and beyond
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916385176682496 |
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| author | Grantham, Jon Granville, Andrew |
| author_facet | Grantham, Jon Granville, Andrew |
| contents | We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes $u_n$, or else there exists a constant $c_u>0$ (which we can give good approximations to) such that there are $\sim c_u \log N$ primes $u_n$ with $n\leq N$, as $N\to \infty$. We compare our conjecture to the limited amount of data that we can compile. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07894 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fibonacci primes, primes of the form $2^n-k$ and beyond Grantham, Jon Granville, Andrew Number Theory 11A41 We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes $u_n$, or else there exists a constant $c_u>0$ (which we can give good approximations to) such that there are $\sim c_u \log N$ primes $u_n$ with $n\leq N$, as $N\to \infty$. We compare our conjecture to the limited amount of data that we can compile. |
| title | Fibonacci primes, primes of the form $2^n-k$ and beyond |
| topic | Number Theory 11A41 |
| url | https://arxiv.org/abs/2307.07894 |