Generalized taxicab number

Short description: Smallest number expressable as the sum of j numbers to the kth power in n ways

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 Taxicab(k, j, n) is the smallest number — if it exists — that can be expressed as the sum of j numbers to the kth positive power in n different ways. For k = 3 and j = 2, 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, Taxicab(5, 2, n) is not known for any n ≥ 2:
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.

The largest variable of $a^5+b^5=c^5+d^5$ must be at least 3450.

See also

• Cabtaxi number

References

• Ekl, Randy L. (1998). "New results in equal sums of like powers". Math. Comp. 67 (223): 1309–1315. doi:10.1090/S0025-5718-98-00979-X. 

External links

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

de:Taxicab-Zahl#Verallgemeinerte Taxicab-Zahl