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Generalized taxicab number

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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., ?

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.

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

Euler showed that

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]

See also

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References

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  1. ^ 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.
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