An Interesting Number
Clive Wilson
April 2005

A famous story among mathematicians tells of the great mathematician G. H. Hardy visiting his famous (but very sadly short-lived) protégé Srinivasar Ramanujan. Hardy remarked that he took note of the number of the taxi in which he had arrived, but unfortunately it appeared to be a rather mundane number: 1729.

Ramanujan instantly replied, "On the contrary, 1729 is a most interesting number. It is the smallest number that can be represented as the sum of two cubes in two different ways!"

I most recently read this story in Dr. Riemann's Zeroes by Karl Sabbagh, an overview of the history and current status of the famous Riemann Hypothesis. As soon as I read the story, I thought, "Yes, of course, well, it's 12³ + 1³, which I know from the Feynman story." In the next paragraph of the book, it turned out that some other mathematician had also had the exact same thought, because he mentioned the coincidence of this number appearing in Feynman's book Surely You're Joking Mr Feynman!.

The "Feynman story" is just one of the fascinating stories in the life of that famous, charismatic, Nobel Prizewinning, bongo drum playing physicist, Richard P. Feynman. Challenged to an arithmetical duel by a Japanese abacus salesman, he is hopelessly beaten by the abacus in addition. He ties at multiplication, though, and is ahead of the abacus man with some long division. Finally the Japanese man shouts out "raios cubicos" (which is "cube roots" in Portugese - they were in Brazil, I think, at the time). Feynman remarks in telling this story, "Cube roots! It's hard to think of a more fundamentally difficult problem in arithmetic. It must have been this guy's most impressive trick in abacusland."

However, fortune smiles on Feynman because the number that is chosen to calculate the cube root of is 1729.03. Feynman, who at Los Alamos frequently had to deal with large quantities of water, knew that one cubic foot is 1728 cubic inches, and that therefore the cube root of 1728 is 12. And with this in mind he was able to pull out 12.002 while the man with the abacus had just got as far as knowing it was 12-something.

So what I'm wondering is, instead of merely being the "smallest number that is the sum of two cubes in two different ways", 1729 should be designated as "the smallest non-trivial number to be mentioned in two different mathematicians' anecdotes"?

I leave to the readers the other means of obtaining 1729 by summing two cubes.

References

1. R. Feynman, Surely You're Joking, Mr. Feynman!, W. W. Norton & Company; Reprint edition (April 1, 1997)
2. K. Sabbagh, Dr. Riemann's Zeroes, Atlantic Books (November 22, 2002)

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