Generalized taxicab number

Unsolved problem in mathematics

Does there exist any number that can be expressed as a sum of two positive fifth powers in at least two different ways, i.e., $a^5+b^5=c^5+d^5$?

More unsolved problems in mathematics

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.

$\begin{array}{rl}\mathrm{T}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{b}(1,2,2)& =4=1+3=2+2\\ \mathrm{T}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{b}(2,2,2)& =50=1^2+7^2=5^2+5^2\\ \mathrm{T}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{b}(3,2,2)& =1729=1^3+12^3=9^3+10^3\end{array}$

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

Euler showed that

$$\mathrm{T}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{b}(4,2,2)=635318657=59^4+158^4=133^4+134^4.$$

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.

See also

• Cabtaxi number

External links

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