Generalized taxicab number

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

${\displaystyle \mathrm {Taxicab} (1,2,2)=4=1+3=2+2.}$

${\displaystyle \mathrm {Taxicab} (2,2,2)=50=1^{2}+7^{2}=5^{2}+5^{2}.}$

${\displaystyle \mathrm {Taxicab} (3,2,2)=1729=1^{3}+12^{3}=9^{3}+10^{3}}$ — famously stated by Ramanujan.

Unsolved problem in mathematics: Does there exist any number that can be expressed as a sum of two positive fifth powers in at least two different ways, i.e., ${\displaystyle a^{5}+b^{5}=c^{5}+d^{5}}$? (more unsolved problems in mathematics)

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 fifth powers in more than one way, and it is not known whether such a number exists.[1]