20141110, 09:47  #1 
(loop (#_fork))
Feb 2006
Cambridge, England
2^{4}·5·79 Posts 
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^5d^5e^5f^5=0 0<=a<=b<=c, 0<=d<=e<=f (a,b,c,d,e,f)=1 c<f<20023 I'm guessing this means that there exists a genuszero 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 20141110 at 09:50 
20141110, 09:56  #2 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
13234_{8} Posts 
What does "(a,b,c,d,e,f)=1" mean?

20141110, 10:52  #3 
(loop (#_fork))
Feb 2006
Cambridge, England
6320_{10} Posts 
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)

20141110, 10:55  #4 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}×1,447 Posts 
I understand. Thanks.
Last fiddled with by retina on 20141110 at 10:56 Reason: If my post number was in hex then this was my 10,000th post. 
20141110, 12:44  #5  
Nov 2003
7232_{10} Posts 
Quote:
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 Eulersum website. 

20141110, 12:49  #6  
Nov 2003
2^{6}·113 Posts 
Quote:


20141111, 14:50  #7 
"William"
May 2003
New Haven
2^{2}×3^{2}×5×13 Posts 

20141111, 15:05  #8 
(loop (#_fork))
Feb 2006
Cambridge, England
2^{4}·5·79 Posts 
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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