About the Number 1729 (One Thousand Seven Hundred Twenty-Nine)

According to G.H. Hardy (English mathematician 01877-01947), the number 1729 is the "smallest number expressible as the sum of two cubes in two different ways."

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

Masahiko Funiwara (Japanese mathematician) showed that "1729 is one of four natural numbers (the others are 81 and 1458 and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversed self, yields the original number."

     1 + 7 + 2 + 9 = 19
         19 * 91 = 1729 
Exercises
  1. Show that Funiwara's algorithm works for the number 1458.

  2. Show that Funiwara's algorithm works for the number 81.

  3. The number 91 is a factor of 1729. 91 is the sum of 4^3 and 3^3. 91 is also the sum of _____^3 and (-5)^3.

Update::2015.10.15

Phys.org::Mathematicians find 'magic key' to drive Ramanujan's taxi-cab number

   "Srinivasa Ramanujan, who had an uncanny sense for the idiosyncratic 
    properties of numbers, somehow knew that 1729 can be represented 
    as 1 cubed + 12 cubed and 9 cubed + 10 cubed, and no smaller 
    positive number can be written in two such ways." -- Phys.org

Creator: Gerald Thurman [gthurman@gmail.com]
Created: 19 December 2007

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