I did find a paper (undated, but with latest reference from 2001) stating that the fifth power problem had been solved by 1964, by Chen Jing-Run: As the result of the work of many mathematicians we now know that g(k) [the number of positive kth powers necessary to represent any positive integer] is determined by: (*) g(k) = 2^k + [(3/2)^k] - 2 provided that (**) 2^k {(3/2)^k} + [(3/2)^k] <= 2^k; if this fails, then g(k) = 2^k + [(3/2)^k] + [(4/3)^k] - theta where theta is 2 or 3 according as the quantity [(4/3)^k] [(3/2)^k] + [(4/3)^k] + [(3/2)^k] equals or exceeds 2^k. It is conjectured that (**) always holds, in which case (*) would always hold. It's known that (**) holds for k <= 471,600,000, and that there are at most a finite number of exceptions. To show that (**) always holds, says this paper, it would suffice to prove that (***) {(3/4)^k} <= 1 - (3/4)^(k-1) . (Where {} is the fractional part, and [] is the greatest integer function.) Hmm, (***) sounds so deceptively simple! --Dan On May 18, 2014, at 10:44 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I've seen various older references (Dickson 1931, Herstein & Kaplansky 1978), citing evidence that every positive integer is the sum of no more than 37 positive 5th powers (with, e.g., 223 requiring that many).
But I haven't seen a claim that this has been proved.
Can anyone please say whether this has been proved or not?*
Thanks,
Dan _________________________________________________________________________ * I'm also curious about the number proposed h(n) by Herstein & Kaplansky as the Waring number for all nth powers -- has this been proved to be the exact maximum number of nth powers needed to represent any positive integer?
But for the moment I'm much more interested in 5th powers.
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