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arXiv · 2307.07894

Fibonacci primes, primes of the form $2^n-k$ and beyond

Jon Grantham·Andrew Granville
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Abstract

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.

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BibTeXRIS

Jon Grantham, Andrew Granville. 2023-07-15. Fibonacci primes, primes of the form $2^n-k$ and beyond. https://doi.org/10.1016/j.jnt.2024.02.002

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