Consider N positive integers which are not necessarily distinct such that their sum of cubes is equal to 2011 (base ten) and their sum of squares is a perfect square.
Determine the smallest value of N for which this is possible. What is the next smallest value of N?
let xk be the number of occurences of the integer k in the set
then we need sum xk*k^3 = 2011 and sum xk*k^2 = s^2 for some integer s. 12<=2011^(1/3)<13 so the largest element possible is 12. This gives for a simple exhaustive search that gives the following three sets of minimal length 10
and the next longest has length 11 with 4 sets
In total, there are 53340 sets
Posted by Daniel
on 2011-08-28 19:06:43