33 = (8,866,128,975,287,528)³ + (-8,778,405,442,862,239)³ + (-2,736,111,468,807,040)³ - Page 2

arbooker · 2019-09-07, 07:06

Quote:

Originally Posted by rudy235

Yet the numbers have increased only by one order of magnitude. (10)

So the question is this. Is this a completely off the chart finding? or are Heath Brown Calculations off base?

"From one solution to the next" means for that specific number. We still only know one solution for 33, and I'm not expecting to see the second one in my lifetime.

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

sweety439 · 2019-09-07, 17:58

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

We only select "primitive solutions", i.e. gcd(x, y, z) must be 1, e.g. for n=24, the solution (x, y, z) = (2, 2, 2) is not allowed, since its gcd is 2 (not 1), thus we select (x, y, z) = (-2901096694, -15550555555, 15584139827), another example is for n=48, the solution (x, y, z) = (-2, -2, 4) is also not allowed, and we select (x, y, z) = (-23, -26, 31)

sweety439 · 2019-09-07, 18:08

Quote:

Originally Posted by sweety439

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

We only select "primitive solutions", i.e. gcd(x, y, z) must be 1, e.g. for n=24, the solution (x, y, z) = (2, 2, 2) is not allowed, since its gcd is 2 (not 1), thus we select (x, y, z) = (-2901096694, -15550555555, 15584139827), another example is for n=48, the solution (x, y, z) = (-2, -2, 4) is also not allowed, and we select (x, y, z) = (-23, -26, 31)

No solutions for n = 4 or 5 mod 9, since cubes are = 0, 1 or 8 mod 9

sweety439 · 2019-09-08, 14:48

Quote:

Originally Posted by sweety439

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

We only select "primitive solutions", i.e. gcd(x, y, z) must be 1, e.g. for n=24, the solution (x, y, z) = (2, 2, 2) is not allowed, since its gcd is 2 (not 1), thus we select (x, y, z) = (-2901096694, -15550555555, 15584139827), another example is for n=48, the solution (x, y, z) = (-2, -2, 4) is also not allowed, and we select (x, y, z) = (-23, -26, 31)

More data:

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

(the smallest n not = 4 or 5 mod 9 without known solution is 114)

sweety439 · 2019-09-08, 16:08

Quote:

Originally Posted by sweety439

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

We only select "primitive solutions", i.e. gcd(x, y, z) must be 1, e.g. for n=24, the solution (x, y, z) = (2, 2, 2) is not allowed, since its gcd is 2 (not 1), thus we select (x, y, z) = (-2901096694, -15550555555, 15584139827), another example is for n=48, the solution (x, y, z) = (-2, -2, 4) is also not allowed, and we select (x, y, z) = (-23, -26, 31)

Conjecture: There are infinitely many such solutions (0 <= |x| <= |y| <= |z|, gcd(x,y,z) = 1, and none of x+y, y+z, z+x is 0) for all n not = 4 or 5 mod 9

sweety439 · 2019-09-08, 19:27

Quote:

Originally Posted by sweety439

More data:

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

(the smallest n not = 4 or 5 mod 9 without known solution is 114)

After 114, the data continues with....

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

fivemack · 2019-09-10, 07:49

Is there an investigation of this using the big algebraic-geometry hammers available anywhere? (For example, can you prove that x^3+y^3+z^3-N=0 does not admit elliptic fibrations, in the way that x^4+y^4+z^4-N*w^4=0 does and often allows you to construct arbitrarily large solutions)

I'm guessing this would be the Heath-Brown 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.

R.D. Silverman · 2019-09-10, 09:45

Quote:

Originally Posted by fivemack

Is there an investigation of this using the big algebraic-geometry hammers available anywhere? (For example, can you prove that x^3+y^3+z^3-N=0 does not admit elliptic fibrations, in the way that x^4+y^4+z^4-N*w^4=0 does and often allows you to construct arbitrarily large solutions)

I'm guessing this would be the Heath-Brown 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.

Also, free generators on elliptic curves tend to be "few and far between" and lifting points
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 sub-exponential
algorithm. It failed precisely because the heights of points blow up as one lifts to Q.

This is quite similar to what you say.

arbooker · 2019-09-10, 12:08

Quote:

Originally Posted by fivemack

I'm guessing this would be the Heath-Brown 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.

I think it's expected that there is no such structure to the solutions here--Noam Elkies would be the person to ask.

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.

bsquared · 2019-09-24, 18:40

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

ATH · 2019-09-25, 16:45

3:02 "...to be honest we have yet to throw anything at The Charity Engine guys, that caused them to break a sweat..."

So lets ask The Charity Engine what the 52nd (known) Mersenne Prime is?

2019-09-07, 17:58	#13
sweety439 Nov 2016 2²×3×5×47 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 We only select "primitive solutions", i.e. gcd(x, y, z) must be 1, e.g. for n=24, the solution (x, y, z) = (2, 2, 2) is not allowed, since its gcd is 2 (not 1), thus we select (x, y, z) = (-2901096694, -15550555555, 15584139827), another example is for n=48, the solution (x, y, z) = (-2, -2, 4) is also not allowed, and we select (x, y, z) = (-23, -26, 31) Last fiddled with by sweety439 on 2019-09-07 at 18:06

2019-09-10, 07:49	#18
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 2×3,191 Posts	Is there an investigation of this using the big algebraic-geometry hammers available anywhere? (For example, can you prove that x^3+y^3+z^3-N=0 does not admit elliptic fibrations, in the way that x^4+y^4+z^4-Nw^4=0 does and often allows you to construct arbitrarily large solutions) I'm guessing this would be the Heath-Brown 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 2019-09-10 at 07:50*

2019-09-24, 18:40	#21
bsquared "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

2019-09-25, 16:45	#22
ATH Einyen Dec 2003 Denmark 101111011101₂ Posts	3:02 "...to be honest we have yet to throw anything at The Charity Engine guys, that caused them to break a sweat..." So lets ask The Charity Engine what the 52nd (known) Mersenne Prime is?