## Archive for **December 20th, 2007**

## An interesting 4-digit number

You know the 4-Digit lottery is big business in Singapore.

Whenever someone buys a new car, he/she is likely to place a bet on the number. Of course, once in a while somebody’s car number will win a substantial prize, and reinforce the hypothesis that a ‘new’ car number is ‘lucky’.

Here is a special 4-digit number ‘1729’. No need to rush out and place your bets though. You may not think it is special but our Mathematics colleagues know it is recorded in mathematical history.

The story goes that when the famous mathematician G.H. Hardy came to visit his sick friend, the Indian mathematical genius Ramanujan, he arrived in a taxi-cab. As a conversation opener, Hardy said to Ramanujan, who loved to work with numbers, that the taxicab he came in had a very dull and most uninteresting number, ‘1729’.

It is reported that Ramanujansaid to Hardy, “On the contrary, 1729 is a very interesting number. It is the smallest number that can be expressed as the sum of two cubes in two different ways”. Well, the two ways of expressing the number as the sum of two cubes are:

1-cube + 12–cube (1 + 1728), and 9-cube + 10-cube (729 + 1000).

How Ramanujan could do all these mental gymnastics was just incredible.

I wonder if Hardy had him what the next number was, that had such properties (being expressed as the sum of two cubes in two different ways), how Ramanujan would have answered him?

*Re-trievia recalled: Excerpts from Year 2002
Tan Wee Kiat*