20190907, 07:06  #12  
"Andrew Booker"
Mar 2013
5·17 Posts 
Quote:
That's not to say that small gaps never happen. To get a feel for it, check out Sander Huisman's data on the solutions for \(\max\{x,y,z\}<10^{15}\): https://arxiv.org/src/1604.07746v1/a...s_20160426.txt 

20190907, 17:58  #13 
Nov 2016
2^{2}×3×5×47 Posts 
The solutions to x^3 + y^3 + z^3 = n (where n is not = 4 or 5 mod 9) with smallest z and smallest y, 0 <= x <= y <= z, gcd(x,y,z) = 1 (i.e. only primitive solutions are selected), and none of x+y, y+z, z+x is 0 (if no this condition, then for every positive cube z^3, all (x, x, z) are solutions, and these solutions are all trivial)
Code:
n, x, y, z 1, 9, 10, 12 2, 0, 1, 1 3, 1, 1, 1 6, 1, 1, 2 7, 0, 1, 2 8, 9, 15, 16 9, 0, 1, 2 10, 1, 1, 2 11, 2, 2, 3 12, 7, 10, 11 15, 1, 2, 2 16, 511, 1609, 1626 17, 1, 2, 2 18, 1, 2, 3 19, 0, 2, 3 20, 1, 2, 3 21, 11, 14, 16 24, 2901096694, 15550555555, 15584139827 25, 1, 1, 3 26, 0, 1, 3 27, 4, 5, 6 28, 0, 1, 3 29, 1, 1, 3 30, 283059965, 2218888517, 2220422932 33, 2736111468807040, 8778405442862239, 8866128975287528 34, 1, 2, 3 35, 0, 2, 3 36, 1, 2, 3 37, 0, 3, 4 38, 1, 3, 4 39, 117367, 134476, 159380 42, 12602123297335631, 80435758145817515, 80538738812075974 43, 2, 2, 3 44, 5, 7, 8 45, 2, 3, 4 46, 2, 3, 3 47, 6, 7, 8 48, 23, 26, 31 51, 602, 659, 796 52, 23961292454, 60702901317, 61922712865 Last fiddled with by sweety439 on 20190907 at 18:06 
20190907, 18:08  #14  
Nov 2016
2^{2}×3×5×47 Posts 
Quote:


20190908, 14:48  #15  
Nov 2016
B04_{16} Posts 
Quote:
Code:
53, 1, 3, 3 54, 7, 11, 12 55, 1, 3, 3 56, 11, 21, 22 57, 1, 2, 4 60, 1, 4, 5 61, 0, 4, 5 62, 2, 3, 3 63, 0, 1, 4 64, 3, 5, 6 65, 0, 1, 4 66, 1, 1, 4 69, 2, 4, 5 70, 11, 20, 21 71, 1, 2, 4 72, 7, 9, 10 73, 1, 2, 4 74, 66229832190556, 283450105697727, 284650292555885 75, 4381159, 435203083, 435203231 78, 26, 53, 55 79, 19, 33, 35 80, 69241, 103532, 112969 81, 10, 17, 18 82, 11, 11, 14 83, 2, 3, 4 84, 8241191, 41531726, 41639611 87, 1972, 4126, 4271 88, 3, 4, 5 89, 6, 6, 7 90, 1, 3, 4 91, 0, 3, 4 92, 1, 3, 4 93, 5, 5, 7 96, 10853, 13139, 15250 97, 1, 3, 5 98, 0, 3, 5 99, 2, 3, 4 100, 3, 6, 7 101, 3, 4, 4 102, 118, 229, 239 105, 4, 7, 8 106, 2, 3, 5 107, 28, 48, 51 108, 948, 1165, 1345 109, 2, 2, 5 110, 109938919, 16540290030, 16540291649 111, 296, 881, 892 

20190908, 16:08  #16  
Nov 2016
5404_{8} Posts 
Quote:
Last fiddled with by sweety439 on 20190908 at 16:09 

20190908, 19:27  #17  
Nov 2016
2^{2}·3·5·47 Posts 
Quote:
Code:
115, 6, 10, 11 116, 1, 2, 5 117, 0, 2, 5 118, 3, 3, 4 119, 2, 6, 7 120, 946, 1531, 1643 123, 1, 1, 5 124, 0, 1, 5 125, 3, 4, 6 126, 0, 1, 5 127, 1, 4, 4 128, 54, 77, 85 129, 1, 4, 4 132, 1, 2, 5 133, 0, 2, 5 134, 1, 2, 5 135, 2, 6, 7 136, 225, 582, 593 137, 9, 11, 13 138, 77, 86, 103 141, 2, 2, 5 142, 3, 7, 8 143, 7023, 84942, 84958 144, 2, 3, 5 145, 7, 8, 10 146, 5, 9, 10 147, 50, 56, 67 150, 260, 317, 367 151, 1, 3, 5 152, 0, 3, 5 153, 1, 3, 5 154, 4, 5, 7 155, 3, 4, 4 156, 2232194323, 68844645625, 68845427846 159, 80, 119, 130 160, 2, 3, 5 161, 2, 7, 8 162, 3, 4, 5 163, 21, 26, 30 164, 45, 47, 58 168, 1, 7, 8 169, 0, 7, 8 170, 1, 7, 8 171, 5, 6, 8 172, 15161, 17044, 20357 173, 14543, 30569, 31629 174, 7, 7, 8 177, 2, 7, 8 178, 10, 13, 15 179, 3, 3, 5 180, 223403, 441721, 460002 181, 2, 4, 5 Last fiddled with by sweety439 on 20190908 at 19:28 

20190910, 07:49  #18 
(loop (#_fork))
Feb 2006
Cambridge, England
2×3,191 Posts 
Is there an investigation of this using the big algebraicgeometry hammers available anywhere? (For example, can you prove that x^3+y^3+z^3N=0 does not admit elliptic fibrations, in the way that x^4+y^4+z^4N*w^4=0 does and often allows you to construct arbitrarily large solutions)
I'm guessing this would be the HeathBrown analysis alluded to at the start of the thread ... it is noticeable from Huisman's tables that the size of successive solutions in cases with multiple solutions goes up in roughly the exponential way you'd expect if you were applying some mapping to points P, 2P, 3P ... on some elliptic curve. Last fiddled with by fivemack on 20190910 at 07:50 
20190910, 09:45  #19  
Nov 2003
2^{2}×5×373 Posts 
Quote:
on a curve over a local field to Q via (say) the Tate pairing causes the heights of the points to "blow up" exponentially. A number of years ago there was an effort to create a 'XEDNI' [reverse of 'INDEX'] attack on ECDLP to yield a subexponential algorithm. It failed precisely because the heights of points blow up as one lifts to Q. This is quite similar to what you say. 

20190910, 12:08  #20  
"Andrew Booker"
Mar 2013
5×17 Posts 
Quote:
We do make use of elliptic curves for part of the computation. You can fiber the solutions over the parameter that I called d in the video on 33. For fixed d the solutions correspond to integral points on a particular elliptic curve (a Mordell curve, in fact). We use that to rule out very small values of d, which would otherwise take a long time or special code to handle. 

20190924, 18:40  #21 
"Ben"
Feb 2007
3,371 Posts 
Congratulations again to arbooker for the discovery of the third representation of 3 as a sum of three cubes!
https://www.youtube.com/watch?v=GXhzZAem7k0 