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Whenever I read the anecdote about Hardy, Ramanujan and the taxi number 1729 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.)

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    $\begingroup$ Since Ramanujan was from India, which presumably used Imperial measures at the time, he would probably have been taught at school that there were $1728$ cubic inches in a cubic foot. He would also have known that $729 = 9^{3}$. So it is perhaps not surprising that he was aware that $1729$ was a sum of cubes in two different ways. There also aren't many positive integer cubes less than $1729$. $\endgroup$
    – Geoff Robinson
    Commented Apr 26, 2016 at 15:30
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    $\begingroup$ I don't know that it was off the top of his head. There is classical number theory (involving Gaussian primes) about the number of ways of a number can be represented as the sum of two squares, and it wouldn't be unnatural to try to investigate the situation about sums of two cubes, so maybe it's just something he happened to know from past investigations. $\endgroup$
    – Todd Trimble
    Commented Apr 26, 2016 at 16:03
  • $\begingroup$ Ok good points -- I guess my astonishment was due to my ignorance (or at least not thinking properly about the issue). $\endgroup$
    – Dominic van der Zypen
    Commented Apr 26, 2016 at 19:12
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    $\begingroup$ It is classical that $r_2(k)$ and $r_3(k)$ can be arbitrary large. It is conjectured (but it is also widely open) that for $n\geq 5$ we have $r_n(k)\leq 1$. I don't know the status of $n=4$ from the top off my head. $\endgroup$
    – GH from MO
    Commented Apr 26, 2016 at 19:41
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    $\begingroup$ From the Scientific American's review of a recent film about Ramanujan: `...why Ramanujan already knew that 1729 was a sum of two cubes in two different ways: he encountered this fact while searching for “near solutions” to the impossible whole number equation $x^3 + y^3 = z^3$. This is revealed at the bottom of this notebook page, which is in Ramanujan’s own handwriting.' And you can see a picture of the relevant part of this notebook in the review. See blogs.scientificamerican.com/guest-blog/… $\endgroup$
    – shane.orourke
    Commented Apr 28, 2016 at 9:23

2 Answers 2

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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
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You may be interested in this article The 1729 $K3$ surface by Ken Ono and Sarah Trebat-Leder, which is aimed at exactly this question of how Ramanujan knew $1729$ so well. Briefly, Ramanujan had studied parameterizations of $a^3+b^3= c^3+ d^3$ in detail, and especially the near misses to Fermat for cubes $a^3 + b^3 =c^3 \pm 1$. The paper by Ono and Trebat-Leder sets this in the context of demonstrating that a certain elliptic curve over ${\Bbb Q}(t)$ has rank $2$.

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