Generalized taxicab number
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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.
It has been shown by Euler that
- Taxicab(4,2,2) = 635318657 = 594 + 1584 = 1334 + 1344
However, Taxicab(5, 2, n) is not known for any n ≥ 2; No positive integer is known at all which can be written as the sum of two fifth powers in more than one way.
