  We define a discrete version of the bilinear spherical maximal function, and
show bilinear $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to
l^{r}(\mathbb{Z}^d)$ bounds for $d \geq 3$, $\frac{1}{p} + \frac{1}{q} \geq
\frac{1}{r}$, $r>\frac{d}{d-2}$ and $p,q\geq 1$. Due to interpolation, the key
estimate is an $l^{p}(\mathbb{Z}^d)\times l^{\infty}(\mathbb{Z}^d) \to
l^{p}(\mathbb{Z}^d)$ bound, which holds when $d \geq 3$, $p>\frac{d}{d-2}$. A
key feature of our argument is the use of the circle method which allows us to
decouple the dimension from the number of functions compared to the work of
Cook.
