Generalized taxicab number

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

<math>\begin{align} \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 \end{align}</math>

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

Euler showed that

<math display=block>\mathrm{Taxicab}(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.</math>

However, Template:Math is not known for any Template:Math:
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.<ref>Template:Cite book </ref>

See also

• Cabtaxi number

References

<references />

• Template:Cite journal

External links

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