OFFSET
1,2
COMMENTS
The only certain values are G(1) = 1, G(2) = 4 and G(4) = 16.
See A002804 for the simpler problem of Waring's original conjecture, which does not restrict the bound to "sufficiently large" numbers. - M. F. Hasler, Jun 29 2014
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 395 (shows G(4) >= 16).
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
LINKS
H. Davenport, On Waring's problem for fourth powers, Annals of Mathematics, 40 (1939), 731-747. (Shows that G(4) <= 16.)
Wikipedia, Waring's Problem.
Trevor D. Wooley, On Waring's problem for intermediate powers, arXiv:1602.03221 [math.NT], 2016.
EXAMPLE
It is known that every sufficiently large number is the sum of 16 fourth powers, and 16 is the smallest number with this property, so a(4) = G(4) = 16. (The numbers 16^k*31 are not the sum of fewer than 16 fourth powers.)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
N. J. A. Sloane, Jan 28 2003; entry revised Jun 29 2014.
STATUS
approved