mersenneforum.org Integer points on a homogeneous senary quintic
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 2014-11-10, 09:47 #1 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 24·5·79 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^5-d^5-e^5-f^5=0 0<=a<=b<=c, 0<=d<=e<=f (a,b,c,d,e,f)=1 c
 2014-11-10, 09:56 #2 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 132348 Posts What does "(a,b,c,d,e,f)=1" mean?
 2014-11-10, 10:52 #3 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 632010 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)
 2014-11-10, 10:55 #4 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 22×1,447 Posts 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.
2014-11-10, 12:44   #5
R.D. Silverman

Nov 2003

723210 Posts

Quote:
 Originally Posted by fivemack 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
(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.

2014-11-10, 12:49   #6
R.D. Silverman

Nov 2003

26·113 Posts

Quote:
 Originally Posted by R.D. Silverman (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/

2014-11-11, 14:50   #7
wblipp

"William"
May 2003
New Haven

22×32×5×13 Posts

Quote:
 Originally Posted by fivemack 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?

 2014-11-11, 15:05 #8 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 24·5·79 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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