Generalized taxicab number

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List of unsolved problems in mathematics
Does there exist any number that can be expressed as a sum of 2 positive 5th powers in at least 2 different ways, i.e., a5 + b5 = c5 + d5?

In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number which 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.

\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 - famously stated 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 which can be written as the sum of two fifth powers in more than one way.[1]

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References[edit]

  1. ^ Guy, Richard K. (2004). Unsolved problems in number theory (third edition). New York, New York, USA: Springer-Science+Business Media, Inc. p. 437. ISBN 0-387-20860-7. 

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