Both 31 and 32 cannot be shown to be a sum of three cubes.
Prove that there is an infinite number of such "twins" (successive integers).
A cube may be {0,1,8}mod3^2.
Hence the sum of 1, 2, or 3 cubes is {0,1,2,3,6,7,8}mod3^2.
In any 9 numbers there will be a successive pair worth {4,5}mod3^2 which cannot be the sum of 3 or less cubes.

Posted by broll
on 20121127 00:34:28 