From: elkies@abel.MATH.HARVARD.EDU (Noam Elkies) Newsgroups: sci.math.numberthy Subject: x^3 + y^3 + z^3 = d Message-ID: <199607092011.QAA13190@ramanujan.math.HARVARD.EDU> Representing an integer d as a sum x^3+y^3+z^3 of three integer cubes is a long-standing problem. It is known that this cannot be done for d congruent to 4,5 mod 9 (clearly) or d=0 (Euler); there are infinitely many polynomial solutions whenever d is a cube, and finitely many when d is twice a cube, e.g. the only known polynomial solution for d=2 is (x,y,z)=(1+6t^3+1,1-6t^3,-6t^2) [with permutations of x,y,z or changes of variable in t consider to be the same]. There are no analytic results for any other d, though the usual heuristics suggest that given d (not 4 or 5 mod 9) solutions should occur infinitely often, but rarely, with asymptotically c*log(N) solutions in |x|,|y|,|z|1) whose only known representations came from representations of d' by scaling, and several d for which only one representation was known, e.g. 12 = 7^3 + 10^3 - 11^3. R.Guy wrote with more recent references: W.Conn and L.N.Vaserstein (in "The Rademacher Legacy to Mathematics", Contemp. Math. 166 (1994)) and D.R.Heath-Brown, W.M.Lioen and H.J.J. te Riele (Math.Comput.1993) found several new solutions, including 2 = 1214928^3 + 3480205^3 - 3528875^3 (the first one not accounted for by the above polynomial) and 39 = 117367^3 + 134476^3 - 159380^3 (39 was the third-smallest unknown value of d, the first two being 30 and 33), and more recently (mid-1995) Richard F. Lukes found several new solutions such as 110 = 109938919^3 + 16540290030^3 - 16540291649^3 with x,y,z rather large but y+z small. The methods were: 1) Extending the exhaustive search for small values of x^3+y^3=z^3, which takes time N^2 (ignoring logarithms) and logarithmic space; 2) For specific d, solve for each potential value of y+z the congruence x^3 == d mod y+z and search over x satisfying this congruence. This takes time N, and can also efficiently find larger solutions such as Lukes' above with y+z small, but only gets solutions for one d at a time and in practice requires more space (to sieve over y+z) and, at least as implement by Heath-Brown et al., auxiliary conditions on the arithmetic of Q(cbrt(d)). Some weeks ago I realized that it is possible to find all solutions of |x^3+y^3+z^3|<<|x|+|y|+|z|<, greg@math.math.ucdavis.edu (Greg Kuperberg) wrote: => => In article <53tf24\$jlp@dfw-ixnews2.ix.netcom.com> fredh@ix.netcom.com => (Fred W. Helenius) writes: => }A perfect cube is congruent to -1, 0, or 1 modulo 9, => }so the sum of three cubes cannot be 4 mod 9. => => In that case, which integers are the sum of three cubes? Nobody knows. It may be that every number not congruent to plus-or-minus 4 mod 9 is a sum of three cubes; on the other hand, no one has ever found such an expression for 30. See Richard K Guy, Unsolved Problems in Number Theory, D5. Gerry Myerson (gerry@mpce.mq.edu.au) ============================================================================== From: ikastan@alumnae.caltech.edu (Ilias Kastanas) Newsgroups: sci.math Subject: Re: x^3+y^3=z^3+w^3 Date: 8 Jan 1997 10:37:35 GMT In article , Dennis Yelle wrote: >In article <5aqjv8\$vij@news.alaska.edu> fthg@aurora.alaska.edu (Hannibal Grubis) writes: >>'m curious about whether or not there is a pattern to numbers >>expressible as the sum of two cubes in two different ways that are NOT of >>the form (12k)^3 + k^3 = (10k)^3 + (9k)^3 >> i.e. Ramanujan's famous number 1729 >>Would it be difficult to write a program to develop a list of these >>numbers? > >Probably quite difficult to write a program to list all of them. > >Here are the ones I found when I limited the search >to those with 1 <= x,y,z,w <= 100: > > 1729 = 1^3 + 12^3 = 9^3 + 10^3 > 4104 = 2^3 + 16^3 = 9^3 + 15^3 > 13832 = 2^3 + 24^3 = 18^3 + 20^3 ... > 1016496 = 47^3 + 97^3 = 66^3 + 90^3 If x, y; s, t are a solution to x^3 + y^3 = s^3 + t^3, then another solution is x(x^3 + 2y^3)(s^3 - t^3), t(t^3 + 2s^3)(x^3 - y^3); y(y^3 + 2x^3)(s^3 - t^3), s(s^3 + 2t^3)(x^3 - y^3). Ilias ============================================================================== From: Don Reble Newsgroup: sci.math Subject: Re: Seeking number theory problems, part II Date: Wed Jul 17 22:58:38 CDT 2002 > Find integers x, y, z such that x^3+y^3+z^3=30 or prove that no such > integers exist. Ok. 30 = (2220422932)^3 + (-2218888517)^3 + (-283059965)^3. I got that from http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/cube01.htm 33 is still unsolved, apparently. -- Don Reble djr@nk.ca ============================================================================== [Note: that web page points to tables, which include the following reference for the case N=30: -- djr] Eric Pine, Kim Yarbrough, Wayne Tarrant and Michael Beck, University of Georgia Noam D. Elkies,Rational points near curves and small nonzero |x3-y2| via lattice reduction, ANTS IV (2000)