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%I #36 Dec 20 2023 13:30:56
%S 1,4,4,16,6,9,8,32,13,12,12,16,14,15,16,64,18,27,20,25
%N 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.
%C The only certain values are G(1) = 1, G(2) = 4 and G(4) = 16.
%C 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
%D 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).
%D 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.
%H H. Davenport, <a href="http://www.jstor.org/stable/1968889">On Waring's problem for fourth powers</a>, Annals of Mathematics, 40 (1939), 731-747. (Shows that G(4) <= 16.)
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Waring%27s_problem">Waring's Problem</a>.
%H Trevor D. Wooley, <a href="http://arxiv.org/abs/1602.03221">On Waring's problem for intermediate powers</a>, arXiv:1602.03221 [math.NT], 2016.
%e 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.)
%Y Cf. A002376, A002377, A002804, A174406.
%K nonn,hard,more
%O 1,2
%A _N. J. A. Sloane_, Jan 28 2003; entry revised Jun 29 2014.
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