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Junpei Fujimoto | KEK | Japan

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S. Ramanujan

S. Ramanujan( from http://www.math.rochester.edu/u/faculty/doug/)

I was asked by a friend of mine to read the blog to give him an example that numbers are interesting. It reminds me a very famous episode of S. Ramanujan, who was an Indian mathematician in the early 20th cent. He found tremendous numbers of mathematical formulas or relationships among numbers. When he was in the bed of a hospital, an English mathematician, G.H. Hardy visited his room, and said that he took a taxi of which plate number was 1729, and that this number was quite trivial one. But Ramanujan immediately answered that 1729 was quite interesting one, because it was the minimum number which could be presented by the sum of two cubic numbers in two ways, as follows;

1729 = 12^3 + 1^3 = 10^3 + 9^3.

It is natural to have a question why Ramanujan so quickly remembered 12^3=1728. Those days, Fermat’s Last Theorem, relating to cubic numbers, was one of the center problems among mathematicians. So it is not so strange, in some sense.


R.P. Feynman( photo by Magnus Waller)

This question is, however, solved by R.P. Feynman who was an American physicist, establishing the theory of electron and photon, quantum electrodynamics(QED) in the middle of 20th cent. Almost the same number appears in the book, ‘Surely you are joking, Mr. Feynman!’ That number is 1729.03.

At a restaurant in Brazil, Feynman had to compete against a Japanese who was very good at counting on the abacus. The problem was to calculate cubic root of 1729.03. Feynman immediately remembered 1728 = 12^3 because a cubic foot is 1728 cubic inches. Then Feynman used Taylor expansion to get better accurate solution, 12.002, before the Japanese got a result with his abacus. We, Japanese, do not use inch-feet system. But we can learn from this story that 1 foot is 12 inches! I wonder a large fraction of Europeans or Americans must know well about 1728=12^3.

But it was just Ramanujan who realized 1729 had such an interesting nature. Especially it is not trivial to prove 1729 is the minimum one. In this context, it is natural to agree with another English mathematician, J.E. Littlewood to say “Every positive integer is one of Ramanujan’s personal friends”.