The Dullness of 1729

One of the best-known anecdotes in the history of mathematics is 
about a visit that Hardy paid to Ramanujan in the hospital in 1917.
The latter had been an obscure young clerk in his native India until 
just a few years earlier, when he had written to Hardy - then the
world's most famous mathematician - asking Hardy to look at some of
his work.  Hardy immediately recognized that the young man had an
extraordinary gift, and arranged for Ramanujan to go to Cambridge 
in 1913.  The work that Ramanujan did there between 1913 and 1918 
is legendary.  Unfortunately, he fell ill in 1917, and thereafter
spent much of his time in the hospital.  He seems to have believed
that his health problems were due to an inability to get suitable
food in England, (he was a strict vegitarian, and cooked all his
food himself), so in 1919 he returned to India.  Alas, his health
did not improve, and he died in 1920.

The famous anecdote is that during one visit to Ramanujan in the
hospital at Putney, Hardy mentioned that the number of the taxi cab
that had brought him was 1729, which, as numbers go, Hardy thought 
was "rather a dull one".  At this, Ramanujan perked up, and said
"No, it is a very interesting number; it is the smallest number
expressible as a sum of two cubes in two different ways."  This was
the sort of thing that prompted Littlewood to say "every positive
integer was one of [Ramanujans'] personal friends".

I was reminded of this story after noticing that, beginning at the 
1729th decimal digit of the transcental number e, the next ten 
successive digits of e are 0719425863.  This is the first appearance
of all ten digits in a row without repititions.  So if anyone ever 
tells me that 1729 is a dull number, I intend to affect a moment of 
contemplation and then say "Not at all, it is the first occurrance 
of all ten digits consecutively in the decimal representation of e".
Now THAT's impressive.

But seriously, it's always seemed implausible to me that Hardy
thought 1729 was a dull number.  In addition to being the smallest 
number that is a sum of two cubes in two distinct ways, it's also a
Carmichael Number, i.e., a pseudoprime relative to EVERY base.

Incidentally, the first three Carmichael Numbers are 561, 1105, and
1729.  Notice that 1105 is expressible as a sum of two SQUARES in
more ways than any smaller number, and of course 561 is expressible 
as a sum of two first-powers in more ways than any smaller number.