2017
^{2017} = 2017*2017
^{2016} = 2017 * (2017
^{672})³
So k is at most 2017
Playing around with x
^{x} for small x the following pattern arises
if x=3n then k=1 suffices since the number is a cube
if x=3n+1 then k<=x
if x=3n+2 then k<=x²
Since 2017 = 3*672+1 it fits the second rule.
Admittedly I only really tried 4
^{4} but trying to subtract cubes without leaving gaps that must be filled with many smaller cubes is difficult.
I might note that with 5
^{5} its very easy to do better than 5
^{2} = 25:
5
^{5} = 13
^{3} + 9
^{3} + 5
^{3} +4
^{3} + 2
^{3} + 1
^{3} + 1
^{3} ^{}

Posted by Jer
on 20150816 08:04:53 