[mkl]

> When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability.

This is the way the story is always presented, and I think that's usually how it's intended, but I think it's quite misleading for another reason too. If you've ever made or looked at a table of cubes, the famous fact really jumps out (in base 10). I'm serious, look:

   n    n³
  --------
   1     1
   2     8
   3    27
   4    64
   5   125
   6   216
   7   343
   8   512
   9   729
  10  1000
  11  1331
  12  1728

The two pairs of cubes are 1000 and 729, and 1728 and 1, and 1000 and 1 make the addition trivial and the similarity obvious (and 729 and 1000 are even right next to each other, one row away from 1728!). With that observation, it doesn't take much effort to try the smaller possibilities and see that 1729 is the smallest number that can be written as the sum of two cubes two different ways. Ramanujan knew numbers and their relationships intimately, better than Hardy, who knew more theory. I think Ramanujan knew the fact about 1729 already, and that you are right about the taxi number coincidence being more surprising and, well, impressive.

(Yes, I've commented on this before: https://news.ycombinator.com/item?id=21165031)