Abstract:
The usual answer is that Waring's problem was solved Hilbert in 1909, but this is an outrageous oversimplification. What Hilbert proved was not what the XVIII century mathematicians asked.
In its original XVIII century form the classical Waring problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers $n$ can be written as sums of s non-negative k-th powers, n=x_1^k+\ldots+x_s^k. The conjectural value of g(s) was predicted in 1772 by Euler jr. For k=3 this problem what indeed solved by Wieferich in 1909, and for k\ge 7 by Dickson and Pillai in 1936. However the solution for the remaining cases n=4, 5, 6 required several more decades, and for the case n=4 explicitly stated by Waring himself in1770 was only completed with the advent of computers in 1984. However, in the XIX century the problem was modified by Jacobi and others as the quest of finding such minimal s=G(k) that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. As of today, in these stronger forms the problem is not solved even for k=3. In the talk I sketch the historical development that led toa complete solution of the classical problem, and also mention some further variations such as the rational Waring problem, the easier Waring problem, etc.
