Ramanujan Number

When you love mathematics you can see magic in numbers. Your face gets lit up when you observe something new about a number. Something similar and very interesting happened with great Indian mathematician

Srinivasa Iyengar Ramanujan

You can see the title of this post is Ramanujan Number. You might have already guessed that he might have a stumbled up on some very interesting number with some peculiar characteristics. If you have guessed that, you are right.  Ramanujan number is 1729.

1729 is also known as the Hardy – Ramanujan number . This number is also called the Taxicab number.

Ramanujan number is so named after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan.

In Hardy's own words:

“I remember once going to see him 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 positive cubes in two different ways."

The numbers is such that,

1729 = 1^3 + 12^3 = 9^3 + 10^3

Some observations related to Ramanujan Number–

1. If negative cubes are allowed, 91 is the smallest possible number with similar quality 91 = 6^3 + (?5)^3 = 4^3 + 3^3
2. Interestingly 91 is also a factor of 1729. (91x19=1729)
3. If taking “positive cubes” would not have been a condition, Ramanujan number could have been ?91, ?189, ?1729, and further negative numbers
4. 1729 is also the third Carmichael number and the first absolute Euler pseudoprime.  (If you want to know more about this numbers I can discuss it in some other post)
5. Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 81, 1458, and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:     1 + 7 + 2 + 9 = 19;                       19 x 91 = 1729
6. Till date only 10 Taxicab numbers are known.  Subsequent Taxicab numbers are found using computers.

Friend’s there are many other properties attached to this great number suggested by various great mathematicians. If you could observe anything special about this number, leave a comment below to delight us.

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