Generalized taxicab number

Smallest number expressable as the sum of j numbers to the kth power in n ways

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.,

a^{5}+b^{5}=c^{5}+d^{5}

(more unsolved problems in mathematics)

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.

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

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

Euler showed that

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

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.^[1]

References

^ Guy, Richard K. (2004). Unsolved Problems in Number Theory (Third ed.). New York, New York, USA: Springer-Science+Business Media, Inc. ISBN 0-387-20860-7.