Wednesday, April 2, 2014

Mystery Number

Find the number $x$ that satisfies these two properties:
  • The digits of $x$ add up to a number $y$ where $x$ equals $y$ times the number you get when you reverse the digits of $y$.
  • Reverse the digits of $x$ and find the prime factors of the number you get. Then take the sum of the squares of these prime factors and halve it. Removing the digit 0 from the new number yields back $x$
Hint : $x$ is a four digit number.


The answer is 1729.  The number is known as the Hardy-Ramanujan number after Ramanujan and the mathematician and Godfrey Hardy. It has another interesting property: you can write it as a sum of cubes in two different ways:
\[  1729 = 1^3+12^3=9^3+10^3. \]
Ramanujan Srinivasa Ramanujan, 1887-1920.

Hardy told the following story: "I remember once going to see [Ramanujan] when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.' "