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| {{unsolved|mathematics| Does there exist any number that can be expressed as a sum of 2 positive ''5''th powers in at least 2 different ways, i.e., ''a''<sup>5</sup> + ''b''<sup>5</sup> <nowiki>=</nowiki> ''c''<sup>5</sup> + ''d''<sup>5</sup>?}}
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| In [[mathematics]], the '''generalized taxicab number''' ''Taxicab''(''k'', ''j'', ''n'') is the smallest number which can be expressed as the sum of ''j'' ''k''th positive powers in ''n'' different ways. For ''k'' = 3 and ''j'' = 2, they coincide with [[Taxicab number]]s.
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| It has been shown by [[Euler]] that
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| :<math>\mathrm{Taxicab}(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.</math>
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| 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.<ref>{{cite book
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| | last = Guy
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| | first = Richard K.
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| | authorlink = Richard K. Guy
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| | title = Unsolved problems in number theory (third edition)
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| | publisher = Springer-Science+Business Media, Inc.
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| | date = 2004
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| | location = New York, New York, USA
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| | pages = 437
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| | url = http://books.google.com/books?id=1AP2CEGxTkgC&printsec=frontcover#v=onepage&q=&f=false
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| | doi =
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| | id =
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| | isbn = 0-387-20860-7 }}
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| </ref>
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| It can be easily verified on a home computer using a simple brute force search that the ''Taxicab''(5, 2, ''2'') problem has no solutions with { ''a, b, c, d'' } all less than 1,000. A more in-depth search shows the same is true for all combinations up to 4,000. A lower bound on the solution is
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| :<math>\mathrm{Taxicab}(5, 2, 2) > 1,024,000,000,000,000,000 = 1.024 * 10^{18}.</math>
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| ==See also==
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| *[[Cabtaxi number]]
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| ==References==
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| <references />
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| ==External links==
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| *[[Walter Schneider (mathematician)|Walter Schneider]]:[http://web.archive.org/web/20050425023736/http://www.wschnei.de/number-theory/taxicab-numbers.html Taxicab numbers]
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| [[Category:Number theory]]
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| [[de:Taxicab-Zahl#Verallgemeinerte Taxicab-Zahl]]
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