# Integers that are a sum of two $k$th powers in $n$ different ways

Do there exist infinitely many $k$ such that for all $n$ we can find a sequence $x_i$ of distinct natural numbers

such that $x_1^k+x_2^k=x_3^k+x_4^k=\cdots=x_{2n-1}^k+x_{2n}^k$ ?

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This is an open problem. Little is known for $k>3$.
In particular, nobody knows if there is a solution of $$x_1^4 + x_2^4 = x_3^4 + x_4^4 = x_5^4 + x_6^4,$$ or $$x_1^5 + x_2^5 = x_3^5 + x_4^5.$$
For details, see Richard K. Guy, Unsolved Problems in Number Theory, section D, in the vicinity of pages 211–216. Guy gives an extensive bibliography, and cites Hardy and Wright for the claim that there are numbers that can be expressed as a sum of two cubes in arbitrarily many ways, although examples with $n=3$ and $n=4$ were not found until 1957 and 1991, respectively.
Jean-Charles Meyrignac's web site includes an enormous table giving, for each $m$ and $k$, the smallest $n$ for which a solution to $$\sum_1^m x_i^k = \sum_1^n y_i^k\tag{\bullet}$$ is known. You want $m=n=2$, but for $k>3$ and $m=2$ the best known $n$ in the table is always greater than 2, and increases rapidly with $k$. The site has extensive information about equation $\bullet$.