Sums of Powers - Ramanujan and his Number 1729

You have reached the web pages of RAJESH RAM (maths@nivedh.com).  This page contains some of the results that I have found   on Sums of Powers relating to an identity of Ramanujan.   Click here for a Text Version of this page.  Click here for my mathematics home page

This page contains an example of a very long and beautiful equality. To see the significance of Ramanujan's Number and one other identity of his in this equality scroll down to the end of the page. 1729 as we all know is 12^3 + 1^3 = 10^3 + 9^3. In the example below, 1729 is contained in a beautiful manner.

See the Text Version of this page to see a general form.

= its not over yet. I will be adding the rest soon.

How is 1729 contained in the above ?

43^2 - 3*40 = 48^2 - 23*25 = 45^2 - 8*37 = 47^2 - 15*32 = 1729

How is one of Ramanujan's Identity contained here ?

Take the first piece :

Take only the Numerators or Denominators and you would get one of Ramanujan's Identity on Sums of Powers. ( You can refer about the below identity in "Ramanujan for Lowbrows", Bruce Berndt, S. Bhargava , Aug-Sep 1993, p 648-649) and I believe this is also the topic of the article "A remarkable identity found in Ramanujan's third Notebook", Glasgow Math. J. 34(1992) 341-345 by the same authors. Please note that I have not seen the second reference - so I could be wrong about it.  But the first one I have seen and it describes in detail about the beauty of the equation below which happens to be a small subset of the one above )

You can interchange (3, 40, 43 ) with (8, 37, 45) and the result will remain unchanged. The same applies to any interchange you make with any of the triples. The equality will still be valid.

Please note that a general solution is posted in the text version of this page.  A more general general solution to this equality will be posted soon.