... Problem and Goldbach’s Conjecture 20-2008 Waring’s Problem Lagrange’s 4 square theorem s that every number can be written as the sumof four integer squares. Waring proposed a generalisation of this theorem in 1770, stating that every natural number is the sum of a fixed number g(k) of kth power integers, where k is any given positive integer and g(k) depends only on k. In this paper, we will investigate what is currently known about this problem.
Warings Problem was proven for all k by Hilbert in 1909 (Ellison 1971), with the Hilbert-Waring Theorem. Prior to that, the problem was solved in the affirmative for specific ks, especially for small k.
If we let "s" stand for the number of kth powers, the...