I am interested in unimodular integral positive-definite quadratic forms taking values in multiples of $m$, for some integer $m \geq 2$, which are maps $Q : \mathbb Z^n \to m \mathbb Z_{\geq 0}$. (Unimodularity means that the lattice $L =(\mathbb Z^n, Q)$ is equal to its dual $L^{\vee}$, or that it has covolume 1).
It is well-known that when $m=2$, there is an even unimodular lattice if and only if the dimension $n$ is divisible by $8$.
What happens if $m \geq 3$ ? Can we find examples for some $n$ at least ?
$Q(x)=b(x,x)$ with the symmetric bilinear form $b(x,y)=\frac{Q(x+y)-Q(x)-Q(y)}2$.
$L=\Bbb{Z}^n$ is unimodular means that the dual lattice $L^{\vee}=\{ y\in \Bbb{R}^n, \forall x\in L, b(x,y)\in \Bbb{Z}\}$ is $L$.
This implies that the $\Bbb{Z}$-ideal generated by the $b(x,y),x,y\in L$ is the whole of $\Bbb{Z}$ so that for some integers $c_{i,j}$ $$1=\sum_{i,j}c_{i,j} b(e_i,e_j)= \sum_{i,j}c_{i,j} \frac{Q(e_i+e_j)-Q(e_i)-Q(e_j)}2$$ And hence $Q(L)$ cannot be contained in $m\Bbb{Z}$ for some $m\ge 3$.