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 = 1 ^{2} + 7 ^{2} = 5 ^{2} + 5 ^{2} \\ Taxicab ( 3 , 2 , 2 ) & = 1729 = 1 ^{3} + 12 ^{3} = 9 ^{3} + 10 ^{3 \end{align}}

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

Euler showed that

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

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

• Cabtaxi number

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.

• Randy L. . Ekl . 10.1090/S0025-5718-98-00979-X . New results in equal sums of like powers. 1998 . Math. Comp.. 67. 223 . 1309–1315. 1474650. free.

External links

• Generalised Taxicab Numbers and Cabtaxi Numbers
• Taxicab Numbers - 4th powers
• Taxicab numbers by Walter Schneider