Whenever I read the [anecdote about Hardy, Ramanujan and the taxi number 1729](https://en.wikipedia.org/wiki/1729_(number)) I'm amazed that it could have occurred to anyone just off the top of their head that 1729 can be written as the sum of two cubes in two different ways -- and that it is the smallest such number.


At all events, there are several ways to look at this in a more general way. For positive integers $n, k$ let us set $$r\_n(k) = | \{(x,y): x\leq y \text{ and } x^n + y^n = k\} |.$$


For what, if any, $n,y\geq 2$ is $r\_n^{-1}(\{y\})$ is infinite? 


(Also partial answers and/or examples are very welcome.)



Generalizing Ramanujan's "1729 story"
There are many articles that study the quantity you call $r\_n(k)$ using sieve methods. Among them I mention the following, which give highly non-trivial bounds for the number of $k<X$ such that $r\_n(k)>1$. If you look at these, and also forward reference them using MathSciNet, you should be able to find the state of the art.


* T.D. Browning, Equal Sums of Two kth Powers, Journal of Number Theory, Volume 96, Issue 2, October 2002, Pages 293–318.
* C. Hooley. On another sieve method and the numbers that are a sum of two hth powers. Proc. London Math. Soc., 226 (1981), pp. 30–87.
* C. Hooley, On another sieve method and the numbers that are a sum of two hth powers: II. Journal für die reine und angewandte Mathematik (1996) Volume: 475, page 55-76


