Generalized taxicab number

In number theory, the generalized taxicab number $Taxicab(k, j, n)$ is the smallest number — if it exists — that can be expressed as the sum of $j$ numbers to the $k$th positive power in $n$ different ways. For $k = 3$ and $j = 2$, they coincide with taxicab number.

$$\begin{align} \mathrm{Taxicab}(1, 2, 2) &= 4 = 1 + 3 = 2 + 2 \\ \mathrm{Taxicab}(2, 2, 2) &= 50 = 1^2 + 7^2 = 5^2 + 5^2 \\ \mathrm{Taxicab}(3, 2, 2) &= 1729 = 1^3 + 12^3 = 9^3 + 10^3 \end{align}$$

The latter example is 1729, as first noted by Ramanujan.

Euler showed that

$$\mathrm{Taxicab}(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.$$

However, $Taxicab(5, 2, n)$ is not known for any $n &ge; 2$: 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.