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Old 2014-11-10, 09:47   #1
fivemack
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Default Integer points on a homogeneous senary quintic

As an attempt to procrastinate usefully, I have enumerated the 4460 integer points on

Code:
a^5+b^5+c^5-d^5-e^5-f^5=0
0<=a<=b<=c, 0<=d<=e<=f
(a,b,c,d,e,f)=1
c<f<20023
After resolving a problem associated with double precision, the number of integer points up to height (maximum entry) looks very like a linear function of H, in fact about 0.225*H. Another observation is that 2632 of the points have a+b+c-d-e-f=0.

I'm guessing this means that there exists a genus-zero curve with a mapping onto the variety; there are obviously lots of trivial ones (eg d,e,f being any permutation of a,b,c), which I hope I've ruled out by the inequalities and the GCD condition.

Any idea how to look explicitly for such a curve?

For pure numerology, here are the points with H<300
Code:
13	51	64	18	44	66
3	54	62	24	28	67
8	62	68	21	43	74
53	72	81	56	67	83
39	92	100	49	75	107
53	90	116	26	85	118
68	106	114	73	96	119
1	89	118	38	47	123
3	97	131	39	56	136
17	95	138	13	35	142
42	129	140	65	94	152
91	94	150	28	32	155
9	131	159	63	67	169
36	140	169	68	137	170
13	159	161	43	109	181
61	129	179	74	113	182
28	167	172	39	142	186
18	152	190	44	55	201
113	145	195	58	101	204
10	183	191	18	31	215
11	183	209	19	168	216
153	157	214	5	145	224
12	122	228	27	106	229
126	208	216	151	166	233
147	218	219	157	193	234
23	184	239	59	139	248
35	125	257	2	97	258
163	185	250	3	121	264
67	227	258	97	181	274
179	259	266	193	229	282
80	219	270	132	154	283
30	179	281	99	105	286
201	219	261	106	137	288
3	215	279	40	168	289
93	259	277	107	229	293
71	249	268	136	158	294

Last fiddled with by fivemack on 2014-11-10 at 09:50
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Old 2014-11-10, 09:56   #2
retina
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What does "(a,b,c,d,e,f)=1" mean?
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Old 2014-11-10, 10:52   #3
fivemack
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No common factor on all of them - the equation is homogeneous, so otherwise you get snowed under with things of the form (2a,2b,2c,2d,2e,2f)
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Old 2014-11-10, 10:55   #4
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I understand. Thanks.

Last fiddled with by retina on 2014-11-10 at 10:56 Reason: If my post number was in hex then this was my 10,000th post.
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Old 2014-11-10, 12:44   #5
R.D. Silverman
 
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Quote:
Originally Posted by fivemack View Post
As an attempt to procrastinate usefully, I have enumerated the 4460 integer points on

Code:
a^5+b^5+c^5-d^5-e^5-f^5=0
0<=a<=b<=c, 0<=d<=e<=f
(a,b,c,d,e,f)=1
c<f<20023
After resolving a problem associated with double precision, the number of integer points up to height (maximum entry) looks very like a linear function of H, in fact about 0.225*H. Another observation is that 2632 of the points have a+b+c-d-e-f=0.

I'm guessing this means that there exists a genus-zero curve with a mapping onto the variety; there are obviously lots of trivial ones (eg d,e,f being any permutation of a,b,c), which I hope I've ruled out by the inequalities and the GCD condition.

Any idea how to look explicitly for such a curve?

For pure numerology, here are the points with H<300
Code:
13	51	64	18	44	66
3	54	62	24	28	67
8	62	68	21	43	74
53	72	81	56	67	83
39	92	100	49	75	107
53	90	116	26	85	118
68	106	114	73	96	119
1	89	118	38	47	123
3	97	131	39	56	136
17	95	138	13	35	142
42	129	140	65	94	152
91	94	150	28	32	155
9	131	159	63	67	169
36	140	169	68	137	170
13	159	161	43	109	181
61	129	179	74	113	182
28	167	172	39	142	186
18	152	190	44	55	201
113	145	195	58	101	204
10	183	191	18	31	215
11	183	209	19	168	216
153	157	214	5	145	224
12	122	228	27	106	229
126	208	216	151	166	233
147	218	219	157	193	234
23	184	239	59	139	248
35	125	257	2	97	258
163	185	250	3	121	264
67	227	258	97	181	274
179	259	266	193	229	282
80	219	270	132	154	283
30	179	281	99	105	286
201	219	261	106	137	288
3	215	279	40	168	289
93	259	277	107	229	293
71	249	268	136	158	294
(a,b,c,d,e,f) does not mean pairwise coprime. How do you exclude a=d, b=e, c=f as solutions?

Not also that 20,000 is a long way from oo. I would not infer too much from a small sample
regarding asymptotic behavior.

Note that there are parametric solutions for sum of 3 cubes = sum of 3 cubes.

I would check Euler-sum website.
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Old 2014-11-10, 12:49   #6
R.D. Silverman
 
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Quote:
Originally Posted by R.D. Silverman View Post
(a,b,c,d,e,f) does not mean pairwise coprime. How do you exclude a=d, b=e, c=f as solutions?

Not also that 20,000 is a long way from oo. I would not infer too much from a small sample
regarding asymptotic behavior.

Note that there are parametric solutions for sum of 3 cubes = sum of 3 cubes.

I would check Euler-sum website.
See: http://euler.free.fr/
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Old 2014-11-11, 14:50   #7
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Quote:
Originally Posted by fivemack View Post
Another observation is that 2632 of the points have a+b+c-d-e-f=0.

For pure numerology, here are the points with H<300
Code:
13	51	64	18	44	66
3	54	62	24	28	67
8	62	68	21	43	74
...
All of these have a+b+c-d-e-f=0 mod 30

Does that apply to the larger set, too?
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Old 2014-11-11, 15:05   #8
fivemack
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Well noticed; but it turns out to be a consequence of x^5==x (mod 30) rather than anything more exciting
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