Generalized taxicab number

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Question dropshade.png Unsolved problem 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?
(more unsolved problems in mathematics)


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.

- famously stated by Ramanujan.

Euler showed that

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 ed.). New York, New York, USA: Springer-Science+Business Media, Inc. ISBN 0-387-20860-7. 

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