  Waring's classical problem deals with expressing every natural number as a
sum of g(k) k-th powers. Recently there has been considerable interest in
similar questions for nonabelian groups, and simple groups in particular. Here
the k-th power word is replaced by an arbitrary nontrivial group word w, and
the goal is to express group elements as short products of values of w.
  We give a best possible and somewhat surprising solution for this Waring type
problem for various finite simple groups, showing that a product of length two
suffices to express all elements. We also show that the set of values of w is
very large, improving various results obtained previously.
  Along the way we also obtain new results of independent interest on character
values and class squares in symmetric groups.
  Our methods involve algebraic geometry, representation theory, probabilistic
arguments, as well as three prime theorems from additive number theory
(approximating Goldbach's Conjecture).
