Generalized taxicab number

{{short description|Smallest number expressable as the sum of j numbers to the kth power in n ways}}

{{unsolved|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., $a^5+b^5=c^5+d^5$?}}

In number theory, the generalized taxicab number {{math|Taxicab(k, j, n)}} is the smallest number — if it exists — that can be expressed as the sum of {{mvar|j}} numbers to the {{mvar|k}}th positive power in {{mvar|n}} different ways. For {{math|1=k = 3}} and {{math|1=j = 2}}, they coincide with taxicab number.

$\backslash begin\{align\}$

\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}

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

Euler showed that

$$\backslash mathrm\{Taxicab\}(4,\; 2,\; 2)\; =\; 635318657\; =\; 59^4\; +\; 158^4\; =\; 133^4\; +\; 134^4.$$

However, {{math|Taxicab(5, 2, n)}} is not known for any {{math|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.{{cite book

| last = Guy

| first = Richard K.

| authorlink = Richard K. Guy

| date=2004

| title=Unsolved Problems in Number Theory

| edition=Third

| publisher=Springer-Science+Business Media, Inc.

| location=New York, New York, USA

| url=https://books.google.com/books?id=1AP2CEGxTkgC

| isbn = 0-387-20860-7}}