Esoteric Dissertations from a One-Track Mind

July 13, 2008

Ramanujan and 1729

Filed under: books, math — Tags: , , — codesmithy @ 11:32 am

I’ve been reading The Oxford Book of Modern Science Writing an anthology put together by Richard Dawkins.  It provides another good reason why life is far more interesting than the human imagination.  In it, there is a piece by C. P. Snow that speaks of relationship between Ramanujan and G. H. Hardy.  Ramanujan died at the age of 32. I can’t help but wonder what discoveries he would have made if his life had not been tragically cut short.

G. H. Hardy deserves credit for being able to recognize genius when it confronted him, especially since some of his colleagues apparently didn’t.  Hardy is said to have made the following comment about Ramanujan’s work: “must be true, because, if they were not true, no one would have the imagination to invent them.”  This dynamic is related by Snow in a anecdote he relays when Hardy visits Ramanujan in the hospital.

Hardy used to visit him, as he lay dying in hospital at Putney.  It was on one of those visits that there happened the incident of the taxicab number.  Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance.  He went into the room where Ramanujan was lying.  Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729.  It seemed to me rather a dull number.’  To which Ramanujan replied: ‘No, Hardy!  No, Hardy!  It is a very interesting number.  It is the smallest number expressible as the sum of two cubes in two different ways.’

That is the exchange as Hardy recorded it.  It must be substantially accurate.  He was the most honest of men; and further, no one could possibly have invented it.

How someone just knows that, I will never know.  But from the informal glancing at Ramanujan’s work, he saw numbers differently.  How much of this was a product of not being substantially formally educated, I don’t know.  I do think that learning in a more open-ended fashion, as Ramanujan did, has benefits.  Hardy made some remarks in a similar vein.

Regardless, I double-checked Ramanujan assertion, although there are a few caveats.  Since we are dealing with cubes, we will limit ourselves to whole numbers.  The sum of cubes that one arrives at 1729 are 1^3 + 12^3 = 1729 and 9^3 + 10^3 = 1729.  1729 turns out to be smallest.  The two distinct sum sequence is as follows.

1 12 9 10 1729
2 16 9 15 4104
2 24 18 20 13832
10 27 19 24 20683

In the context of the book, it is meant to show how people of a scientific persuasion see the world differently.  Where one person sees something bland, another sees it as a source of wonder.  In a certain sense, every number has something special about it.  Ramanujan’s gift was seeing what that special thing was.

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