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Positive Perfect Cube Sum (Posted on 2015-08-15) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts An upper bound | Comment 1 of 3
20172017 = 2017*20172016 = 2017 * (2017672
So k is at most 2017

Playing around with xx 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
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