“Generalized taxicab number”的意思、由来-开放百科全书

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Generalized taxicab number

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{{unsolved|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.,

?}}

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]

References

1. ^{{cite book | last = Guy | first = Richard K. | authorlink = Richard K. Guy | date=2004 | title=Unsolved Problems in Number Theory | edition=Third | publisher=Springer-Science+Business Media, Inc. | location=New York, New York, USA | url=https://books.google.com/books?id=1AP2CEGxTkgC&printsec=frontcover#v=onepage&q=&f=false | isbn = 0-387-20860-7}}

{{cite article|first1=Randy L. |last1=Ekl | doi=10.1090/S0025-5718-98-00979-X |

 title=New results in equal sums of like powers|year =1998 | journal=Math. Comp.| volume=67|pages=1309-1315|mr=1474650}}

External links

[https://web.archive.org/web/20150513224105/http://homepage.ntlworld.com/duncan-moore/taxicab/ Generalised Taxicab Numbers and Cabtaxi Numbers]
Taxicab Numbers - 4th powers
[https://web.archive.org/web/20050425023736/http://www.wschnei.de/number-theory/taxicab-numbers.html Taxicab numbers] by Walter Schneider

Taxicab-Zahl#Verallgemeinerte Taxicab-Zahl

1 : Number theory

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