This is from June. I assume the "almost every" is in the 99.999% sense. --Rich ------ On Waring's Problem: Two Cubes and Two Minicubes: Siu-lun Alan Lee We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta}$, as long as $\theta\geq192/869$. An asymptotic formula for the number of such representations is established when $1/4<\theta<1/3$. http://arxiv.org/abs/1006.5142 --- Quoting Dan Asimov <dasimov@earthlink.net>:
Ah, yes, heard of that.
According to Wikipedia: The only difference is that Waring asked about the smallest number g(p) of summed pth powers that can represent *all* integers K, not just sufficiently large ones. Hardy and Littlewood then defined G(p) as the smallest number of pth powers adequate to represent all *sufficiently large* integers.
In case anyone's interested, here's what Wikipedia says:
<< G(3) is at least four (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3 × 109, 1290740 is the last to require six cubes, and the number of numbers between N and 2N requiring five cubes drops off with increasing N at sufficient speed to have people believe G(3)=4; the largest number now known not to be a sum of four cubes is 7373170279850, and the authors give reasonable arguments there that this may be the largest possible.
13792 is the largest number to require seventeen fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 that every number between 13793 and 10245 required at most sixteen, and Kawada, Wooley and Deshouillers extended Davenport's 1939 result to show that every number above 10220 required no more than sixteen). Sixteen fourth powers are always needed to write a number of the form 31·16n.
617597724 is the last number less than 1.3 × 109 which requires ten fifth powers, and 51033617 the last number less than 1.3 × 109 which requires eleven.
Using his improved Hardy-Littlewood method, I. M. Vinogradov has shown that
G(k) <= p*(3*log(p) + 11)
Using his p-adic form of the Hardy-Littlewood-Ramanujan-Vinogradov method A.A. Karatsuba proved the following new bound for:
G(p) < 2p*log(p) + 2p*loglog(p) + 12p .
Also:
<< 4 ≤ G(2) ≤ 4 4 ≤ G(3) ≤ 7 16 ≤ G(4) ≤ 16 6 ≤ G(5) ≤ 17 9 ≤ G(6) ≤ 21 8 ≤ G(7) ≤ 33 32 ≤ G(8) ≤ 42 13 ≤ G(9) ≤ 50 12 ≤ G(10) ≤ 59 12 ≤ G(11) ≤ 67 16 ≤ G(12) ≤ 76 14 ≤ G(13) ≤ 84 15 ≤ G(14) ≤ 92 16 ≤ G(15) ≤ 100 64 ≤ G(16) ≤ 109 18 ≤ G(17) ≤ 117 27 ≤ G(18) ≤ 125 20 ≤ G(19) ≤ 134 25 ≤ G(20) ≤ 142
More at < http://en.wikipedia.org/wiki/Waring's_problem >.
--Dan
Allan wrote: << Affirmative. Your conjecture is Waring's Problem, 1770. Proved by Hilbert, 1905. It's finding M and N for a given p that is really hard.
I wrote: << Does this imply the possibility that:
For every integer p > 0 there exists a number N(p) of pth powers, and a minimum M(p) > 0, such that every integer K > M(p) is the sum of N(p) nonnegative pth powers.
?
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