Do any Taxicab(5, 2, n) exist for n > 1

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For positive integers $k,j,n$, $\operatorname{Taxicab}(k,j,n)$ denotes the least positive integer, if any, expressible as a sum of $j$ positive integer $k$th powers in $n$ distinct ways, counting permutations of summands as the same way. For two fifth powers, sort each pair: $1\le a\le b$, $1\le c\le d$.

Statement recorded in the supplied source.

Does there exist any $\operatorname{Taxicab}(5,2,n)$ with $n>1$? Equivalently, do there exist positive integers $a\le b$, $c\le d$ with $(a,b)\ne(c,d)$ and $a^5+b^5=c^5+d^5$? [S2]

Short English statement

Can a positive integer be represented as a sum of two positive fifth powers in two different ways?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400029 (AMR-093-0029). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, Generalized taxicab number, revision 1373501556 (2026) Lead and fifth-power discussion. https://en.wikipedia.org/w/index.php?title=Generalized+taxicab+number&oldid=1373501556.