All numbers of form $10^{k} + 1$ are composite for $k{\gt}2$ proof
The problem whether or not $G_n=10^{2^n}+1$ is composite for all $n\ge 3$ is open as well as the problem whether or not the Fermat numbers $F_n=2^{2^n}+1$ are composite for all $n\ge 5$, see here, and the article on generalized Fermat numbers.