Trivially, a regular $0$-simplex (point) and $1$-simplex (line segment) can have integer vertices in $0$ and $1$ dimensional Euclidean space respectively. On the other hand, a regular $2$-simplex (equilateral triangle) cannot have vertices in $\mathbb Z^2$.
Is it true more generally that an $n$-simplex cannot have vertices in $\mathbb Z^n$ for $n > 1$? Is there an easy way to see this?
Eugen J. Ionascu provides an example of a regular tetrahedron with vertices in $\mathbb Z^3$: $(0,0,4),(7,0,3),(3,5,0),(4,5,7)$
Do such regular $n$-simplices exist for all $n \neq 2$?