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A higher-rank Mersenne problem. (English) Zbl 1071.11072

Fieker, Claus (ed.) et al., Algorithmic number theory. 5th international symposium, ANTS-V, Sydney, Australia, July 7–12, 2002. Proceedings. Berlin: Springer (ISBN 3-540-43863-7). Lect. Notes Comput. Sci. 2369, 95-107 (2002).
The {Mersenne problem} is the one of estimating the density of primes \[ \mu(X) = \# \{ p \leq X : M_p = 2^p-1 \text{ is (a Mersenne-) prime} \}. \] Some heuristic arguments suggest \(\mu(X) \sim \frac{e^{\gamma}}{\log(2)} \cdot \log \log (X) \), but it is not even known if \(\mu(X)\) diverges with \(X \rightarrow \infty\). As it often happens, an unsolved problem calls for generalizations, and this is what the authors consider in this interesting paper.
A divisibility sequence \(u_n\) is simply one in which \(m | n \Rightarrow u_m | u_n\) and the authors define some {algebraic dynamical systems} giving raise to divisibility sequences; an interesting example are the denominators of the (even) Bernoulli numbers. Asking what the density of primes in a divisibility sequence is, naturally generalizes the Mersenne problem.
Algebraic dynamical systems of rank \(1\) are given by a compact group endomorphism \(T : X \rightarrow X\) and a sequence \(u_n = \# \{ x \in X : T^n(x) = x \}\). For dynamical systems of higher ranks (\(d > 1\)), one considers \(\mathbb Z^d\)-actions of \(d\) commuting automorphisms of a compact abelian group \(X\). The study of periodicity of such systems is complicated and the authors consider the {very} simple case with \(d = 2\) given by \(u_{m,n} = 2^m 3^n - 1\) and the generalized Mersenne density function \[ \mu_2(X) = \# \{ m, n \leq X : u_{m, n} \text{ is a prime} \}. \] They investigate this density function both heuristically and by computations and compare the results of the two investigations. The same is done then for the natural elliptic curve analog of this function.
For the entire collection see [Zbl 0992.00024].

MSC:

11Y11 Primality
11N05 Distribution of primes
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