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.