[math-fun] Unsolved number theory problem
Which integers S are the sum of three cubes: S = J^3 + K^3 + L^3 ??? (None of the form S == ±4 (mod 9), for instance.) Some low numbers are hard to find the J, K, L for, like: 30 = (−662037799708799)3 + 1908092688412843 + 6567116892545653 . The first S for which a solution is unknown is 33. The complete list of S < 1000 with no known solution (as far as I can tell) is as follows: 33 42 114 165 390 579 627 633 732 795 906 921 975 . —Dan
I have spent considerable time looking into this and similar problems over the last couple of months. And think I should be able to give a short summary of the situation. There is a good basic introduction to the problem on Numberphile: https://www.youtube.com/watch?v=wymmCdLdPvM I think the best way to attack the problem computationaly is by using Noam Elkies method as done by Jahnel & al. : http://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-02168-6/S0025-571... This methods finds all "small" solutions via lattice reduction in time N (the upper search bound) More recently Sander G. Huisman has expanded the bound of the search to 10^15, but I suppose he is still using Jahnels old code: https://arxiv.org/abs/1604.07746 There are also ways to look for specific solutions, ex : http://www.ams.org/journals/mcom/2007-76-259/S0025-5718-07-01947-3/S0025-571... I have written code to do searches for specific solutions. But I still feel it is a bit hairy, especially the extraction of cube roots in finite fields, which is often claimed to be easy, but in practice it requires a hodgepodge of techniques from factorization, sieving, code for simple cases, finding roots of unity, Hensel lifting and for instance Tonelli-Shanks algorithms. My code works well enough to find solutions for related problems like I did here: http://www.maroon.dti.ne.jp/fermat/fourcubes1.html but is still work in progress. If I get to the point where I am happy with the code, it could for instance be be purposed for making a distributed BOINC style search effort for specific solutions such as 33 & 42 - but I am not quite there yet. /f On Fri, May 18, 2018 at 10:24 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Which integers S are the sum of three cubes:
S = J^3 + K^3 + L^3
???
(None of the form
S == ±4 (mod 9),
for instance.)
Some low numbers are hard to find the J, K, L for, like:
30 = (−662037799708799)3 + 1908092688412843 + 6567116892545653 .
The first S for which a solution is unknown is 33. The complete list of S < 1000 with no known solution (as far as I can tell) is as follows:
33 42 114 165 390 579 627 633 732 795 906 921 975 .
—Dan
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Dan Asimov -
Frank Stevenson