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 mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number — if it exists — that 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 that can be written as the sum of two fifth powers in more than one way, and it is not known whether such a number exists.

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. MR 1474650.

External links

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