Waring's Problem - The Number g(k)

The Number g(k)

For every k, we denote by g(k) the minimum number s of kth powers needed to represent all integers. Note we have g(1) = 1. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth-powers; these examples show that g(2) ≥ 4, g(3) ≥ 9, and g(4) ≥ 19. Waring conjectured that these values were in fact the best possible.

Lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares; since three squares are not enough, this theorem establishes g(2) = 4. Lagrange's four-square theorem was conjectured in Bachet's 1621 edition of Diophantus; Fermat claimed to have a proof, but did not publish it.

Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that g(4) is at most 53. Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers.

That g(3) = 9 was established from 1909 to 1912 by Wieferich and A. J. Kempner, g(4) = 19 in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, g(5) = 37 in 1964 by Chen Jingrun, and g(6) = 73 in 1940 by Pillai.

Let and {x} denote the integral and fractional part of x respectively. Since 2k-1<3k only 2k and 1k can be used to represent this number and the most economical representation requires -1 2ks and 2k-1 1ks it follows that g(k) is at least as large as 2k + − 2. J. A. Euler, the son of Leonard Euler, conjectured about 1772 that, in fact, g(k) = 2k + − 2. Later work by Dickson, Pillai, Rubugunday, Niven and many others have proved that

g(k) = 2k + − 2 if 2k{(3/2)k} + ≤ 2k

g(k) = 2k + + − 2 if 2k{(3/2)k} + > 2k and + + = 2k

g(k) = 2k + + − 3 if 2k{(3/2)k} + > 2k and + + > 2k.

No values of k are known for which 2k{(3/2)k} + > 2k, Mahler has proved there can only be a finite number of such k and Kubina and Wunderlich have shown that any such k must satisfy k > 471,600,000. Thus it is conjectured that this never happens, i.e. that g(k) = 2k + − 2 for each positive integer k.

The first few values of g(k) are:

1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055 ... (sequence A002804 in OEIS).