At first I thought the comments saying "this is not a proof" were being sarcastic, but I guess readers are simply not aware.

Ramanajuan derived a lot of his mathematical writings from intuition and seldom had proofs to the level of Western expectations.

It's nice that his "lost" formulae have been verified, but more important would be an understanding of the intuition behind them.

I'm hoping that would lead to tools for lighter-weight proofs than the recent heavy-weight and impenetrable Weil et al proofs.

That he seldom had proofs to the level of Western expectations, whatever that means, is an odd assumption to make. You should have read the (captivating) paper that was on the front page two days ago.

"The notebooks contain almost no proofs. Perhaps there are about 10-20 results for which Ramanujan sketches a proof, often only with one sentence. There are several reasons for the absence of proofs.

1. Ramanujan was probably influenced by the style of Carr's book [his primary source for learning mathematics in his youth].

2. Like most Indian students in his time, Ramanujan worked primarily on a slate. Paper was very expensive. Thus, after rubbing out his proofs with his sleeve, Ramanujan recorded only the final results in his notebooks.

3. Ramanujan never intended that his notebooks be made available to the mathematical public. They were his personal compilation of what he had discovered. If someone had asked him how to prove a particular result in the notebooks, undoubtedly Ramanujan could supply a proof.

[...]

It should be emphasized that Ramanujan doubtless thought like any other mathematician; he just thought with more insight than most of us."

- Bruce C. Berndt, An Overview of Ramanujan's Notebooks

http://www.math.uiuc.edu/~berndt/articles/aachen.pdf

I don't really understand this article. It reads like pop science because of all the elided details, but no layperson could possibly understand it. But I'm not sure how much an expert would really out of it either. The stuff about "having solutions for a particular function" is just nonsense.

Also, can anyone actually decipher the handwritten stuff?

Here's their method:

>First, calculate a numerical value for the point of interest. Second, conjecture a closed algebraic form for this number. Third, express the algebraic number as nested radicals. Finally, check the conjectured form with many digits of accuracy.

But this isn't a proof. It's just empirical evidence. At least it seems to me - I'm no mathematician.

Keep reading. "An actual proof can be accomplished using modular equations. "

That's an outline of the proof so mathematicians in the field don't need to explore other nonrelated methods to reach the proof for themselves.