A079611 Waring's problem: conjectured values for G(n), the smallest number m such that every sufficiently large number is the sum of at most m n-th powers of positive integers.

1, 4, 4, 16, 6, 9, 8, 32, 13, 12, 12, 16, 14, 15, 16, 64, 18, 27, 20, 25

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

Table of n, a(n) for n=1..20.

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

Cf. A002376, A002377, A002804, A174406.

Sequence in context: A375745 A289625 A040004 * A246763 A319070 A227074

Adjacent sequences: A079608 A079609 A079610 * A079612 A079613 A079614

Keyword

nonn,hard,more

Author

N. J. A. Sloane, Jan 28 2003; entry revised Jun 29 2014.

Status

approved