Theoretical Statistics and Mathematics Unit, ISI Delhi
Representing (non-negative) integers as sums of powers of integers is a very old question with several aspects, some being more than 2,000 years old. We shall concentrate on some questions of the Waring’s type
(*) n = x_1^s + …. + x_k^s, with 1 < s < k+1.
The classical Waring’s problem (end of XVIII th century) is to determine the smallest k, denoted as g(s), such that every integer n as a representation (*). L. Dickson and S. S. Pillai independently expressed g(s) in terms of s for all s > 6, and Pillai showed that g(6)=73.
In our lecture, we shall then explore questions related to small value of k, essentially k=s and k=s+1, where little is known but where probability theory leads to convincing models.