Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes

The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if (Formula presented.) is a subfield of a local field of characteristic (Formula presented.), then the special upper triangular group (Formula presented.) is minimal precisely when the special linear group (Formula presented.) is. We provide criteria for the minimality (and total minimality) of (Formula presented.) and (Formula presented.) where (Formula presented.) is a subfield of (Formula presented.). Let (Formula presented.) and (Formula presented.) 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 (Formula presented.) : (Formula presented.) is finite; (Formula presented.) is minimal, where (Formula presented.) is the Gaussian rational field; and (Formula presented.) is minimal. Similarly, denote by (Formula presented.) and (Formula presented.) the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let (Formula presented.) Then the following conditions are equivalent: (Formula presented.) is finite; (Formula presented.) is minimal; and (Formula presented.) is minimal.