PROBLEM. *Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms.*

It suffices to show that the terms of the sequence $$\,b_n=\mathrm{e}^n\,\mathrm{mod}\, 2,\,\,\,n\in\mathbb N,$$ are dense in $[0,2]$.

Unfortunately, Weyl's Theorem does not look helpful in this case.

EDIT. As Chris Culter said, the claim that *the terms of the sequence
$$\,b_n=\mathrm{e}^n\,\mathrm{mod}\, 2,\,\,\,n\in\mathbb N,$$
are dense in $[0,2]$* is (or might be) an open problem. Nevertheless, this does not imply that the claim that *the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms* is necessarily an open problem as well. It is also noteworthy that it is relatively easy to construct an irrational $\alpha$ with the property that the sequence $\,\alpha^n\,\mathrm{mod}\, 2,\,\,n\in\mathbb N,$ is NOT dense in $[0,2]$.