Mon October 31, 2011 By:

how many natural numbers are there with the property that they can be expressed as the sum of the cubes of two natural numbers in two different ways ?

Expert Reply
Tue November 08, 2011
Answer - If you ask that any number has to be represented as the sum of cubes of 2 natural numbers in exactly 2 different ways, then there is only 1 such number and it was discovered by a famous mathmatician Ramanujan by accident when he hired a taxi whose number was 1729. Then later on theories on such numbers were developed and these numbers are now known as "taxicab numbers". 
Let us define a number T(n) where n is the different number of ways in which T(n) can be expressed as a sum of cubes of different natural numbers.
In this question of yours, you are asking only two different ways, so n is 2.
And the number is 1729 which is the  (1) sum of cube of 1 and 12 & (2) sum of cube of 9 and 10.
If you put the value of n as 3, it means there are three combinations of these sums of cubes and then the number will be 87539319
For the case of two different ways it is 1729 known yet
Thank you 
Ask the Expert