Generalized taxicab number

In number theory, the generalized taxicab number is the smallest number — if it exists — that can be expressed as the sum of numbers to the th positive power in different ways. For and, they coincide with the taxicab number.
The latter example is 1729, as first noted by Ramanujan.
Euler showed that
However, is not known for any :
No positive integer is known that can be written as the sum of two 5th powers in more than one way, and it is not known whether such a number exists.