The taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in two different ways, and it is equal to $1729=12^3+1^3=10^3+9^3$. There are generalizations to more than two ways, to allowing negative cubes (then the answer is $91=4^3+3^3=6^3-5^3$), and to different powers. In the latter case, the answer is $50=7^2+1^2=5^2+5^2$ for squares, $635318657 = 134^4 + 133^4 = 158^4 + 59^4$ for fourth powers, while for fifth powers it is not known whether any such number exists, that is, whether equation $x^5+y^5=z^5+t^5$ has any non-obvious integer solutions.

It is natural to ask the same question for rational powers, that is, what is the smallest positive integer that can be expressed as a sum of two **rational** $d$-th powers in two different ways. For squares, the answer is $1=(3/5)^2+(4/5)^2=(5/13)^2+(12/13)^2$, see a comment of Wojowu. For cubes, the answer is
$$
6 = \left(\frac{37}{21}\right)^3 + \left(\frac{17}{21}\right)^3 = \left(\frac{2237723}{960540}\right)^3 + \left(-\frac{1805723}{960540}\right)^3.
$$
In fact, $6$ has infinitely many such representations because $x^3+y^3=6z^3$ defines an elliptic curve of rank $1$. For fifth powers, no such positive integers are known, because if they were, we could obtain non-trivial integer solution to $x^5+y^5=z^5+t^5$ after multiplying by denominators.

This leaves the case of $d=4$. That is: what is the smallest positive integer that can be expressed as a sum of two rational fourth powers in two different ways?

The sequence of positive integers $n$ that are the sums of two rational fourth powers starts with $1,2,16,17,32,81,82,\dots$. From the solutions to well-known equations $x^4+y^4=z^4$ and $x^4+y^4=2z^4$ we conclude that such representations for $1,2,16,32,81$ are unique. Jeremy Rouse informed me that in 2001, Flynn and Wetherell proved that the representations of $17$ is unique as well. Hence, the first case for which I currently do not know the answer is $n=82$.

The challenge is that the existence of one representation implies that some standard methods (such as local obstructions, reduction to rank $0$ elliptic curves, or the Mordell-Weil Sieve in its simplest form) does not help for proving the non-existence of a second representation.

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