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 March 21st, 2017, 12:00 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 422 Thanks: 27 Math Focus: Number theory Integer coordinates in a sphere Given a sphere of integer radius R, how many integer coordinates (a, b, c) at most does it contain? Thanks from agentredlum
 March 21st, 2017, 12:09 PM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Do you want to include the points on the surface as well? Thanks from Loren
 March 21st, 2017, 01:07 PM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 422 Thanks: 27 Math Focus: Number theory Yes. please.
 March 21st, 2017, 01:39 PM #4 Senior Member     Joined: Sep 2015 From: USA Posts: 2,460 Thanks: 1340 It's looking like $f(r) = 0.573291 r^{2.98494}$ is a pretty good fit Thanks from Loren
 March 21st, 2017, 02:31 PM #5 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 422 Thanks: 27 Math Focus: Number theory Many thanks, romsek, for the computation. Can this problem be solved analytically?
March 22nd, 2017, 02:33 AM   #6
Math Team

Joined: Jul 2011
From: North America, 42nd parallel

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Quote:
 Originally Posted by Loren Many thanks, romsek, for the computation. Can this problem be solved analytically?
I would suggest a geometric approach that would use a 3d imaging program with every triple integer lattice highlighted as a dot and start with a sphere of radius 1 centered at the origin. Count the dots enclosed by the sphere , increase the radius to 2 , count the dots enclosed by the sphere , repeat a few times and check the numbers to see if you can find a constant ratio. That would at least give you an insight as to what you are up against. A good idea is to write code that counts the dots for you then visually confirm that the code is working properly by comparing it to your count. If you can do all that then you can make predictions for random radii and see how your predictions compare to the program answers.

 March 31st, 2017, 12:03 AM #7 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 422 Thanks: 27 Math Focus: Number theory Ball from cubes Can a ball be constructed out of finite cubes? I once read about a proof of this in the affirmative.

 Tags coordinates, grid, integer, points, sphere

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