1,729 is the least integer which equals the sum of two positive cubes in two different ways (see "taxicab number" on the web):
1,729 = 12^3+1^3 and 1,729 = 10^3+9^3.

a) Find an integer which equals the sum of two positive cubes in THREE different ways.
There is more than one solution.

b) Erasing the word "positive" in the 1st statement allows a lesser positive integer(s??) number(s??) to be presented as a sum of two cubes in two different ways.
Find it (them??).

(In reply to computer solution by Justin)

"Part B: Any solution for this part must have a sum of 2 positive cubes and a difference of 2 positive cubes (same as adding a negative cube) equaling the same value. "

This is not the case as, for example, 9^3 - 2^3 = 16^3 - 15^3 = 721.  Also, for 728, there's a third way besides the two you list, and that's 12^3 - 10^3, and thus a second difference of cubes for this number.

Posted by Charlie on 2010-08-02 16:08:54