If we let "s" stand for the number of kth powers, then g(k) is the least such "s" powers. Some examples of g(k) are: g(1) = 1; g(2) = 4, since from Lagranges 4-square theorem, every natural number is the sum of atleast 4 squares. In addition it was found that 7 requires 4 squares and 23 requires 9 cubes.
Progress was made on Warings Problem by establishing bounds, or the maximum number of powers. For instance, Liouville found that g(4) is at most 53. The work of Hardy and Littlewood also led to other bounds; in particular, they found the upper bound for g(k) to be O(k2k+1).
The work of Hardy and Littlewood also led to the realization that the number G(k) is more fundamental than g(k). Here, G(k) is the least positive integer s such that every sufficiently large integer (greater than some constant) is a sum of at most s kth powers of positive integers. A formula for the exact value of G(k) for all k has not been found, but there have been many bounds established.
1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, 2102137, 4201783, 8399828, 16794048, 33579681, 67146738, 134274541, 268520676, 536998744, 1073933573, 2147771272
Hilberts proof of Warings Problem for all positive k can be seen as proving an equivalent theorem: There are positive integers A and M and positive rationals 1, ..., M, depending only on k, such that each integer N A can be written in the form
Many generalizations of Waring’s Problem have been made. For instance, there is the prime Waring’s problem, and generalizations of the problem to algebraic number fields and arbitrary fields. The problem known as the “easier” Waring’s Problem takes the integer n to be a sequence of numbers x, each to the kth power. All of these variations have led to a Mathematics Subject Classification 11P05 entitled “Waring’s Problem and variants.”
In 1742, Goldbach suggested that every