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March 21st, 2017, 12:00 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 458 Thanks: 29 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?

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?

March 21st, 2017, 01:07 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 458 Thanks: 29 Math Focus: Number theory 
Yes. please.

March 21st, 2017, 01:39 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,550 Thanks: 1401 
It's looking like $f(r) = 0.573291 r^{2.98494}$ is a pretty good fit 
March 21st, 2017, 02:31 PM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 458 Thanks: 29 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 Posts: 3,372 Thanks: 233  Quote:
 
March 31st, 2017, 12:03 AM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 458 Thanks: 29 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. 

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coordinates, grid, integer, points, sphere 
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