Question

what are hardy ramanujan's numbers ???
A short note on that pls !!!

Answer 1

1729 is the smallest number which can be expressed as the cube of two positive integers in two different ways :

1729 can be written as 1³ + 12³ or as 9³ + 10³

The number gets its name from the incident took place when British Mathematician G.H Hardy went to visit his friend Srinivasa Ramanujan in hospital :

Here is there conversation by G.H Hardy :

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 unfavourable 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."

Answer 2

The number is so called because of an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation as follows:

   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 unfavourable 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.

The two different ways are:

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

Numbers of the kind a^3 + b^3 = m^3 + n^3 = x^3 + y^3 are now known as Ramanujan Triples. The corresponding problem is known as TaxiCab(3).

Some observations related to Ramanujan Number–

If negative cubes are allowed, 91 is the smallest possible number with similar quality 91 = 6^3 + (?5)^3 = 4^3 + 3^3

   Interestingly 91 is also a factor of 1729. (91×19=1729)

   If taking “positive cubes” would not have been a condition, Ramanujan number could have been ?91, ?189, ?1729, and further negative numbers

   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)

   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

   Till date only 10 Taxicab numbers are known.  Subsequent Taxicab numbers are found using computers.