Generalized taxicab number
From Wikipedia, the free encyclopedia
| 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 which can be written as the sum of two fifth powers in more than one way.[1]
[edit] See also
[edit] References
- ^ Guy, Richard K. (2004). Unsolved problems in number theory (third edition). New York, New York, USA: Springer-Science+Business Media, Inc.. pp. 437. ISBN 0-387-20860-7. http://books.google.com/books?id=1AP2CEGxTkgC&printsec=frontcover#v=onepage&q=&f=false.
