In 1770, Waring asserted, without proof, that for every fixed positive 
integer r, there is  a positive integer s such that every sufficiently 
large  positive integer can be represented as a sum of s  r^th powers (for 
instance, every positive integer is a sum of four squares, 7 cubes and so 
on).  Waring's assertion was proved in 1920's by Hilbert, Hardy-Littlewood, 
and many others.

We investigate the following question, raised by Nathanson in 1980: Does 
one really need to use all r^{th} in the representations ?
In this talk, we  shall present a nearly optimal answer to this question. 
In fact, we shall answer a much stronger form of the question.