Minimality conditions equivalent to the finitude of Fermat and Mersenne primes

Abstract

It is still open whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning the Mersenne numbers is also unsolved. Extending some results from [9], we characterizethe the Fermat primes and the Mersenne primes in terms of topological minimality of some matrix groups. This is done by showing, among other things, that if $\Bbb{F}$ is a subfield of a local field of characteristic $\neq 2,$ then the special upper triangular group $\operatorname{ST^+}(n,\Bbb{F})$ is minimal precisely when the special linear group $\operatorname{SL}(n,\Bbb{F})$ is. We provide criteria for the minimality (and total minimality) of $\operatorname{SL}(n,\Bbb{F})$ and $\operatorname{ST^+}(n,\Bbb{F}),$ where $\Bbb{F}$ is a subfield of $\Bbb{C}.$ Let $\mathcal F_\pi$ and $\mathcal F_c $ be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for $\mathcal{A}\in \{\mathcal F_\pi, \mathcal F_c\}:$ $\bullet \ \mathcal{A}$ is finite; $\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{SL}(F_n-1, \Bbb{Q}(i))$ is minimal, where $\Bbb{Q}(i)$ is the Gaussian rational field; $\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{ST^+}(F_n-1, \Bbb{Q}(i))$ is minimal. Similarly, denote by $\mathcal M_\pi$ and $\mathcal M_c $ the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let $\mathcal{B}\in\{ \mathcal M_\pi, \mathcal M_c\}.$ Then the following conditions are equivalent: $\bullet \ \mathcal B$ is finite; $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{SL}(M_p+1, \Bbb{Q}(i))$ is minimal; $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{ST^+}(M_p+1, \Bbb{Q}(i))$ is minimal.