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. The 
first three Carmichael Numbers are 561, 1105, and 1729.  (It's
interesting 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.)

Incidentally, Hardy told the story about 1729 as part of his answer
to the question of "whether [Ramanujan's] methods differed in kind
from those of other mathematicians; whether there was anything really
abnormal in his mode of thought." Hardy's answer was that, although
Ramanujan's memory and powers of calculation were very unusual, they
could not reasonably be called "abnormal". He (Hardy) believed that
"all mathematicians think, at bottom, in the same kind of way, and
that Ramanujan was no exception". To illustrate this point, he then
told the story about 1729, but significantly he says that after
hearing Ramanujan's observation on 1729 he (naturally) asked if he 
knew of any number expressible as a sum of two fourth powers in more 
than one way. "[Ramanujan] replied, after a moment's thought, that he 
could see no obvious example, and thought that the first such number 
must be very large." Hardy didn't mention, but quite possibly knew
by the time he wrote about this incident, that Euler had found (more
than a century before) an infinite family of such numbers, the first 
of which is 

        635318657  =  133^4 + 134^4  =  158^4 + 59^4

So, although the story of 1729 is usually presented as an example
of Ramanujan's unusual prowess, it seems that Hardy intended the
overall anecdote to show not only Ramanujan's capabilities, but also
his limitations. Had Ramanujan pondered for a moment and announced
the number 635318657, presumably Hardy would have been ready to
describe his mode of thought as being "abnormal". 

The number 1729 also appears in Richard Feynman's collection of
anecdotes (Surely You're Joking, Mr. Feynman!). In a chapter 
entitled "Lucky Numbers" he tells of going into a small restaurant
in Brazil to eat lunch. He's the only customer, so he has four 
waiters standing around him. Then a Japanese man enters the restaurant,
and he is selling abacuses. The man challenges the waiters to an adding 
contest, but they don't want to lose face, so they tell him to go 
challenge the customer sitting there (Feynman). They first have an 
addition contest, and the abacus wins easily. Then they try multiplication,
and the abacus wins again, but it's a bit closer. Then they try long 
division, and this time it's a tie. As Feynman says, the more difficult 
the problem, the better he can do with pencil and paper compared with 
the abacus. Finally the Japanese man calls out "Raios cubicos!"... he 
wants to challenge Feynman to cube roots. 

Feynman says the man wrote a number, "any old number", down on a piece 
of paper, and he still remembers the number... 1729.03. The salesman 
begins working furiously on his abacus, but Feynman just sits there 
smiling, and says "12.002...". The abacus salesman was beaten, and 
left the restaurant in disgust. The waiters are amazed at Feynman's 
calculating prowess. He explains that he happenned to remember that 
there are 1728 cubic inches in a cubic foot, so the cube root of 1729 
must be just slightly greater than 12. Then he just needed to account 
for the extra 1.03. To do this he neglected the 0.03 and used the 
binomial expansion

            1/3                  1/3
  (1728 + 1)    =  12(1 + 1/1728)    =  12(1 + (1/3)(1/1728) + ...

so the amount by which the cube root of 1729 exceeds 12 is about
4/1728 = 1/432. You can only get two 432's out of 1000, so the first
non-zero digit is 2, leaving a remainder of 136, and bringing down
another zero we know there are three 432's in 1360, so the next
digit is 3, and so on. This gives 12.0023...  

Feynman's book contains many stories similar to this, in which he is 
able to perform a seemingly difficult mental feat, and he then explains
that it was only by luck that he happenned to recognize something about
the problem that made it simple. He says Hans Bethe was an even better
mental calculator, and by similar means. Feynman says "It was easy
for him - every number was near something he knew".