How many integer coordinated points are on the surface of a four-dimensional sphere of radius 85?
The problem's question is: How many integer coordinated points are on the surface of a four-dimensional sphere of radius 85, given that the center of that sphere is at a integer coordinated point itself?
Note By integer coordinated point, a point whose four coordinate values are all integers, e.g., {0,0,0,0} or {-1,17,-42,10^100}.is meant. Real or complex values that are not also integers are excluded.
The answer is 76136.
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1 solution
This is a brute force solution! Using 7.6 seconds of CPU time and 8GB of memory on a 8 core 4.57GHz machine using Mathematica 12 to do the search gives the 76136 answer. The algorithm can be programmed easily in most languages.
Timing [ r = Flatten [ Block [ { v = Range [ − 8 5 , 8 5 ] } , ParallelTable [ If [ w 2 + x 2 + y 2 + z 2 = 7 2 2 5 , { x , y , z , w } , Nothing ] , { x , v } , { y , v } , { z , v } , { w , v } ] ] , 3 ] ]
The length of the r table is 76136.