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It is still open whether there exist infinitely many Fermat primes orinfinitely many composite Fermat numbers. The same question concerning theMersenne numbers is also unsolved. Extending some results from [9], wecharacterizethe the Fermat primes and the Mersenne primes in terms oftopological minimality of some matrix groups. This is done by showing, amongother things, that if $\Bbb{F}$ is a subfield of a local field ofcharacteristic $\neq 2,$ then the special upper triangular group$\operatorname{ST^+}(n,\Bbb{F})$ is minimal precisely when the special lineargroup $\operatorname{SL}(n,\Bbb{F})$ 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 theset of composite Fermat numbers, respectively. As our main result, we provethat the following conditions are equivalent for $\mathcal{A}\in \{\mathcalF_\pi, \mathcal F_c\}:$  $\bullet \ \mathcal{A}$ is finite;  $\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{SL}(F_n-1, \Bbb{Q}(i))$ isminimal, 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 Mersenneprimes and the set of composite Mersenne numbers, respectively, and let$\mathcal{B}\in\{ \mathcal M_\pi, \mathcal M_c\}.$ Then the followingconditions are equivalent:  $\bullet \ \mathcal B$ is finite;  $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{SL}(M_p+1, \Bbb{Q}(i))$ isminimal;  $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{ST^+}(M_p+1, \Bbb{Q}(i))$is minimal.

Minimality conditions equivalent to the finitude of Fermat and Mersenne primes