  The Burau representation is a natural action of the braid group B_n on the
free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to
determine for which values of n this representation is faithful. It is known to
be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long
and Paton improved on Moody's techniques to bring this down to n>5. Their
construction uses a simple closed curve on the 6-punctured disc with certain
homological properties. In this paper we give such a curve on the 5-punctured
disc, thus proving that the Burau representation is not faithful for n>4.
