Generalized taxicab number
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Unsolved problem in mathematics: Does there exist any number that can be expressed as a sum of two positive fifth powers in at least two different ways, i.e., [math]\displaystyle{ a^5+b^5=c^5+d^5 }[/math]? (more unsolved problems in mathematics)
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In number theory, the generalized taxicab number Taxicab(k, j, n) is the smallest number — if it exists — that can be expressed as the sum of j numbers to the kth positive power in n different ways. For k = 3 and j = 2, they coincide with taxicab number.
[math]\displaystyle{ \begin{align} \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 \end{align} }[/math]
The latter example is 1729, as first noted by Ramanujan.
Euler showed that
[math]\displaystyle{ \mathrm{Taxicab}(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4. }[/math]
However, Taxicab(5, 2, n) is not known for any n ≥ 2:
No positive integer is known that can be written as the sum of two 5th powers in more than one way, and it is not known whether such a number exists.[1]
The largest variable of [math]\displaystyle{ a^5+b^5=c^5+d^5 }[/math] must be at least 3450.[citation needed]
See also
References
- ↑ 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. https://books.google.com/books?id=1AP2CEGxTkgC.
- Ekl, Randy L. (1998). "New results in equal sums of like powers". Math. Comp. 67 (223): 1309–1315. doi:10.1090/S0025-5718-98-00979-X.
External links
- Generalised Taxicab Numbers and Cabtaxi Numbers
- Taxicab Numbers - 4th powers
- Taxicab numbers by Walter Schneider
de:Taxicab-Zahl#Verallgemeinerte Taxicab-Zahl
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