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 kth positive power in n different ways.
For k = 3 and j = 2, they coincide with taxicab number.
a x i c a b
( 1 , 2 , 2 )
a x i c a b
{\displaystyle {\begin{aligned}\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{aligned}}}
The latter example is 1729, as first noted by Ramanujan.
Euler showed that
{\displaystyle \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 ≥ 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.