Musings around the "easier" Waring problem

The talk is dedicated to the history of the easier Waring problem and further related problems. Recall, that this problem consists in finding for each natural $k$ the such smallest $s = v(k)$ that all natural numbers $n$ can be expressed as sums of $k$-th powers of integers $n = ±x1^k± ... ± xs^k$ with signs, where the number of summands equals $s$. The history of this problem illustrates lack of continuity and missed opportunities. Officially, it was stated in 1933-34 by Vesely and Wright, but in 1936 Bell observed that the cheap version of this problem (the analogue of Hilbert's result, mere existence of $s$) could have been solved any time after 1851 wheт Tardy has written his identities. I claim that it was precisely the work of Tardy and the conflict with Libri that induced Liouville's interest in the classical Waring problem, which eventually led to its complete solution. In turn, the "easier" Waring problem itself turned out to be terribly much harder than the original one, and is not solved in a single non-trivial case. It turned out to be intimately connected to many deep problems of arithmetic and diophantine geometry. We discuss various aspects of this problem, and further related problems, (rational Waring problem, Waring problems for finite fields, polynomials, number fields, etc.), as well as the connection with the Waring problem at zero (Fermat problem, Euler conjecture, etc.) The two further amazing aspects are the recent computer calculations connected with the representations of specific small numbers as sums of three cubes, etc. (Mordell's challenge), and the contribution of many amateurs (Frolov, Verebrusov, and others) at the early stage.