Work on this problem began in around 2003. I was searching the internet for some results pertaining to the golden ratio, Lucas numbers, and Fibonacci numbers, and I encountered some Fibonacci number work done by Raj Ram. This Fibonacci number work was interesting and I noticed some other interesting things on Raj's pages. In particular the Ramanujan number 1729 material drew my attention. When I initially saw these fraction results (enumerations) I thought they were amazing. Now, a few years later, I still think they are amazing. This prompted me to starting working on this problem.

Initially I thought this must have been some sort of "trick problem" and I thought that if I worked on it for five minutes that I would immediately see the trick. Well, as it turns out, this problem is kind of addictive. You work on it a bit, and it leads you a bit further, and so you keep working on it. Well, I have worked on this problem (and some related problems) for over a year now, and have found many interesing things, but to date still not have found "the trick" that one suspects must exist.

Soon after beginning work on this problem and after finding some initial results, I contacted Raj Ram. We had some discussions, and since then, having done a lot of work on this problem, he has served as mentor on this problem, and has given many suggestions.

Most of the results in these web pages, in particular the linear and quadratic permutation studies, were done in the time frame of around April 2003 to Dec. 2003 (approx.).

Many of these results have been published on the web before. Some of this work appeared on the FEREGO.com web site (where George worked for a while). Portions of this work therefore have appeared on the internet at various times, on and off, since 2003. This particular web page, a main discussion page, is new however.

This has prompted an interest in Ramanujan himself. Since 2003 work on this problem has slowed somewhat. Now I am writing this in 2005.

The summary of results on this page generally are not in chronological order.

The initial set of results obtained, though somewhat simple looking back, were exciting. They are not listed at this time. (But may appear in the future when I get more time to gather materials.)

My original purpose in working on this problem, which we are calling the Ramamurthi Fractions Problem, was to work with Raj to try to systematically write up and publish the incredible results he found (and that are listed at a link given above).

There are at least three interesting aspects to this problem:

• The results posted at Raj's site above are interesting.
• The relationship to the Ramanujan 6-8-10 formula is interesting. His results appear to be deeper than the 6-8-10 formula itself.
• Extensions of Raj's work, namely some permutation studies, and searches for other similar results, is interesting.

Some results are given below.

This pdf write up gives some motivation for Raj's forms. Raj's work, and the forms he poses, are a kind of weak generalization of the incredible Ramanujan 6-10-8 formula.

The weak form of Ramanurthi's fractions are in the sense given in this paper.

This paper tabulates results but is still unsatisfactory in many ways. Ramamurthi's forms are motivated as seen from coming from a manipulation based on the 6-8-10 formula, but after that, we just state results that are true and that can be verified. There is still very little insight into where these other forms come from.

Work on this Ramamurthi fractions problem has rather mistakenly resulted in, as a by-product, a proof of the 6-8-10 formula of Ramanujan given in this pdf.

The proof was "accidental" in that we never set out to find a proof of it. Indeed, I never imagined that I would ever be involved in any sort of proof of the 6-8-10 formula whatsoever. (Mathematica actually found the factorization.) So I am rather surprised, as you are.

The factorization can be observed, in simpler form, in the paper in #1 above. However that coordinate system is not fully general, so the factorization there gives a proof only on a nonlinear manifold of lower dimension. The current paper in #2 generalizes the coordinate transformation so that it is fully general. It is a one-to-one coordinate transform.

This proof was found by generalizing the factorization given in the previous pdf (#1 above). This was before the author inspected the proof of the 6-8-10 formula given in the Berndt series Vol. 4, p. 102, and before any of the other cited references were studied. Our proof here is different, we believe, but is based on using a similar factorization of each of the Fn functions for n=6, 8, and 10. Berndt's proof also uses factorizations for the proof we note. The proofs differ in that we prove the factorizations by showing common roots. Berndt shows the factorizations via other very different means. We are able to show that the 6, 8, and 10 forms each respectively have 6, 8, and 10 roots, and this is sufficient to prove the factorization by showing roots. Berndt uses fewer roots in his proof. We also note that the factorizations given by us are larger and deeper factorizations than those noted and used by Berndt, et. al.

A lot of discussion gets skipped, and I am jumping to some results.

Raj Ram (Rajesh Ramamurthi) suggested that permutations of the P, Q, R, and S functions, which he defines, also have many interesting properties. So I began looking for results similar to what Raj found in his web pages using permutations of the P, Q, R, and S functions. Raj knew permutations "also worked" but I am not sure if he had systematically tried to tally the results. I did try to systematically find and record permutation results. I have only explored a few permutations pertaining to a few of the forms. I believe that many more solutions exist.

• Results for a linear permutation problem.
• Results for a quadratic permutation problem.

We note that more permuations problems exist. We have only worked a few. We believe that interesting cubic, quartic, and higher results might exist, but we do not have sufficient computing power to explore them.

We also have not explored all of the forms. In particular the forms containing division have not been combinatorially explored.

As mentioned, this problem has prompted greater interest in Ramanujan. Ramanujan was an incredibly interesting fellow, and a separate page devoted to him discusses interesting aspects.

This page is not done. More items, more discussion, and more links should be added over time.

Often one works on a problem and ends up reinventing the wheel. I do expect to become embarassed at some point regarding this problem since this is not an unusual occurrence.

An interesting web aspect of this problem is how to format a set of output from a computer algebra system. It was easiest to have the algebraic computation system produce two output XML tags: "section" and "p". Then later (on a linux machine) Gnu XSLT templates were used to generate HTML. This allowed the algebra programming to be seperated from the different problem of HTML rendition.

Ironically, even though we have worked together much, I have never met Rajesh Ramamurthi in person. Our collaborations and communications have been purely internet (and email) based.

I conclude by noting that appropriate literature background work so far has been inadequate. Work on obtaining needed background material is currently underway.

Some pertinent references are given in this section. The references given mostly pertain to Ramanujan since this is the main inspiration for the work discussed on this page. References are for books and other items pertaining to Ramanujan and for history and biographical discussions of Ramanujan.

These references give some history information pertaining to Ramanujan:

Learning Ramanujan from the internet is difficult. There are web pages that give summaries of formulas but to actually learn Ramanujan from the web is not recommended. The Math World resource contains many excellent technical summaries of Ramanujan's work, and contains many book references as well.

Credit: some of the CSS style sheets for this page were based on some work and writing that George Schils did when doing some writing for SMV, Jr. Magazine for FEREGO.

Some gnu tools were used in producing some of the renditions.

The computer algebra system mentioned was Mathematica.