  We find a lower bound to the size of finite groups detecting a given word in
the free group, more precisely we construct a word w_n of length n in
non-abelian free groups with the property that w_n is the identity on all
finite quotients of size ~ n^{2/3} or less. This improves on a previous result
of Bou-Rabee and McReynolds quantifying the lower bound of the residual
finiteness of free groups.
