Generalized taxicab number explained

In number theory, the generalized taxicab number is the smallest number — if it exists — that can be expressed as the sum of numbers to the th positive power in different ways. For and, they coincide with taxicab number.

\begin{align} Taxicab(1,2,2)&=4=1+3=2+2\\ Taxicab(2,2,2)&=50=12+72=52+52\\ Taxicab(3,2,2)&=1729=13+123=93+103 \end{align}

The latter example is 1729, as first noted by Ramanujan.

Euler showed that

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

However, is not known for any :
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]

See also

References

  1. Book: Guy , Richard K. . Richard K. Guy

    . Richard K. Guy . 2004 . Unsolved Problems in Number Theory . Third . Springer-Science+Business Media, Inc. . New York, New York, USA . 0-387-20860-7.

External links