1729, and related questions I just read this paragraph: (written by G. H. Hardy, on Ramanujan)

I remember once going to see him when he was lying ill at Putney. I
  had ridden in taxi cab number 1729 and remarked that the number seemed
  to me rather a dull one, and that I hoped it was not an unfavorable
  omen. ‘No,’ he replied, ‘it is a very interesting number; it is the
  smallest number expressible as the sum of two cubes in two different
  ways.’

Was Ramanujan right?
What are other numbers having such property (expressible as the sum of two cubes in two different ways)?
Are there infinite number of them?
And, on the other hand:
What if the word "cubes" is replaced by "5-degree power"? Would such numbers exist? If yes, what would be the smallest?

Another SO question related to 1729: Proof that 1729 is the smallest taxicab number

Very late for this party, but yes, there is an infinite number of taxicab numbers. The complete solution in positive integers to,
$$x_1^3+x_2^3 = x_3^3+x_4^3$$
was given by Choudhry's On Equal Sums of Cubes (1998). For positive integers $a,b,c$,
$$\begin{aligned}
d\,x_1 &= (a^2 + a b + b^2)^2 + (2a + b)c^3\\
d\,x_2 &= (-a^3 + b^3 + c^3)c\\
d\,x_3 &= (a^2 + a b + b^2)^2 - (a - b)c^3\\
d\,x_4 &= (a^3 + (a + b)^3 + c^3)c\end{aligned}$$
where,
$$(a^3-b^3)^{1/3}<\,c\,<\frac{(a^3-b^3)^{2/3}}{a-b}$$
and $d=1$, or chosen such that $\text{GCD}(a,b,c)=1$.
P.S. For Choudhry's complete solution in positive integers to $x_1^3+x_2^3+ x_3^3=x_4^3$, see this post.
