In mathematics, the generalized taxicab number ''Taxicab''(''k'', ''j'', ''n'') is the smallest number — if it exists — that 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 In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), also called the ''n''th Hardy–Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two ''positive'' integer cubes in ...

s. :

\mathrm(1, 2, 2) = 4 = 1 + 3 = 2 + 2.

\mathrm(2, 2, 2) = 50 = 1^2 + 7^2 = 5^2 + 5^2.

\mathrm(3, 2, 2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3

— famously stated by Ramanujan.

Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

showed that :

\mathrm(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.

However, ''Taxicab''(5, 2, ''n'') is not known for any ''n'' ≥ 2:
No positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

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. The largest variable of

\mathrm a^5+b^5=c^5+d^5

must be at least 3450.

References

*{{cite journal, 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, issue=223 , pages=1309–1315, mr=1474650, doi-access=free

External links

Generalised Taxicab Numbers and Cabtaxi Numbers
by Walter Schneider Number theory de:Taxicab-Zahl#Verallgemeinerte Taxicab-Zahl

See also

References

External links