Determine the smallest value of k ≥ 2 such that 2017²⁰¹⁷ is the sum of precisely k positive perfect cubes and prove that no smaller value of k is in conformity with the conditions of the problem.

2017²⁰¹⁷ = 2017*2017²⁰¹⁶ = 2017 * (2017⁶⁷²)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 4⁴ but trying to subtract cubes without leaving gaps that must be filled with many smaller cubes is difficult.


I might note that with 5⁵ its very easy to do better than 5² = 25:
5⁵ = 13³ + 9³ + 5³ +4³ + 2³ + 1³ + 1³

Posted by Jer on 2015-08-16 08:04:53