Generalized taxicab number

Unsolved problem in mathematics: Does there exist any number that can be expressed as a sum of 2 positive 5th powers in at least 2 different ways, i.e., a^5 + b^5 = c^5 + d^5? (more unsolved problems in mathematics)

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.

$\mathrm{Taxicab}(1, 2, 2) = 4 = 1 + 3 = 2 + 2.$

$\mathrm{Taxicab}(2, 2, 2) = 50 = 1^2 + 7^2 = 5^2 + 5^2.$

$\mathrm{Taxicab}(3, 2, 2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3$ - famously stated by Ramanujan.

Euler showed that

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

However, Taxicab(5, 2, n) is not known for any n ≥ 2; no positive integer is known which can be written as the sum of two fifth powers in more than one way.^[1]

See also

• Cabtaxi number

References

1. 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.

External links

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