About sums of three cubes, an unsolved problem: 3x3 semi-magic square of (positive) cubes. Here is a semi-magic cube of 8 cubes out of 9: 51^3 619^3 165^3 618^3 162^3 115^3 178^3 72^3 235788435 www.multimagie.com/English/Enigmas.htm Christian. -----Message d'origine----- De : math-fun <math-fun-bounces@mailman.xmission.com> De la part de Dan Asimov Envoyé : dimanche 20 mai 2018 01:30 À : math-fun@mailman.xmission.com; frankstevensonmobile@gmail.com Objet : Re: [math-fun] Unsolved number theory problem Interesting — thanks, Frank! I wonder if there are large additional families of solutions of X^3 + Y^3 + Z^3 = K if the ring were expanded from the integers to some nice familiar unique factorization domain, like the Gaussian integers Z[i] or Eisenstein integers Z[exp(2πi/3)], or even the Hurwitz integral quaternions.* —Dan ————— * These are the quaternions that are integer linear combinations of the 24 unit quaternions {±1, ±i, ±j, ±k, (±1 ± i ± j ± k)/2}. These 24 form a group under quaternion multiplication known as the binary tetrahedral group and denoted by 2T. The group 2T can be seen as the inverse image p^(-1)(T), under the double-covering map p : S^3 —> SO_3 from the 3-sphere of unit quaternions to the rotation group SO_3 of 3-space R^3, of the 12 elements of the rotation group T of the regular tetrahedron. I.e., 2T = p^(-1)(T). ----- Frank Stevenson wrote: ----- 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 ----- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun