It is well known that 1729 is the smallest integer that can be expressed in two ways as the sum of two perfect cubes: 1729=1³+12³ =9³+10³.

What is the smallest integer that can be expressed in two different ways as the sum of three perfect cubes? And in three different ways as the sum of two perfect cubes?

For this I used Mathematica to generate a list of all the sums of 3 cubes less than or equal to 20.  Then I searched for the smallest sum that occured twice.

5^3+5^3+1^3=6^3+3^3+2^3=251

now for the second part I expanded my search limit to 100 with no result, so as of right now I started the search allowing it to go all the way up to 1000 and am now going to bed, hopefully it will have an answer by the time I wake up :-) 

Posted by Daniel on 2007-06-02 15:49:05