Mixed Tate motives over $\mathbb{Z}$

We prove that the category of mixed Tate motives over $\mathbb{Z}$ is spanned by the motivic fundamental group of ${\text{Pro}}^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\text{Q}$-linear combination of $\zeta (n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_r)$ where $n_i\in \{2,3\}$.