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 $Taxicab(k, j, n)$ is the smallest number — if it exists — that can be expressed as the sum of $j$ numbers to the $k$ th positive power in $n$ different ways. For $k = 3$ and $j = 2$ , they coincide with taxicab number.

${\begin{aligned}\mathrm {Taxicab} (1,2,2)&=4=1+3=2+2\\\mathrm {Taxicab} (2,2,2)&=50=1^{2}+7^{2}=5^{2}+5^{2}\\\mathrm {Taxicab} (3,2,2)&=1729=1^{3}+12^{3}=9^{3}+10^{3}\end{aligned}}$

The latter example is 1729, as first noted 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 $n \geq 2$ :
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.^[1]

The largest variable of $a^{5}+b^{5}=c^{5}+d^{5}$ must be at least 3450.^{[citation needed]}

References[edit]

^ 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.

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[edit]

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

[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.

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