On the Burau representation of $B_3$ modulo $p$

We present an algorithm that, given a prime $p$ as input, determines whether or not the Burau representation of the 3-strand braid group modulo $p$ is faithful. We also prove that the representation is indeed faithful when $p\le 13$. Additionally, we re-pose Salter's question on the Burau representation of $B_3$ over finite fields $\mathbb{F}_p$, and solve it for every $p$.