Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, Hadad, and Kaminer - The Ramanujan machine: Automatically generated conjectures on fundamental constants.
Have these conjectures been proven in the meantime? Are there any partial results?
The two formulas in the abstract were proven by relatively simple methods in a couple of days after the paper appeared on arxiv. See https://arxiv.org/abs/1907.05563 The rest of the formulas inside the paper have not been proven so far, to the best of my knowledge.
Already many people pointed, but the proof of the first identity of the abstract is the following in short.
$3+\displaystyle \frac{-1}{ \displaystyle 4+ \frac{-2}{ \displaystyle 5+ \frac{-3}{ \displaystyle 6+ \frac{-4}{ 7+ \cdots}}}}$ $=\lim_{n\rightarrow\infty}\lim_{x\rightarrow\infty}3+\displaystyle \frac{-1}{ \displaystyle 4+ \frac{-2}{ \displaystyle 5+ \frac{-3}{ \displaystyle 6+ \frac{-4}{ \cdots (n+2)-\frac{n}{x}}}}}$ $\displaystyle=\lim_{n\rightarrow\infty}\lim_{x\rightarrow\infty}(\frac{1}{n!}\frac{x-1}{nx+1-n}+\sum_{k=0}^{n}\frac{1}{k!}) $
$=e$
In the year or so since this question first appeared on MO, there has been further progress on proving some of these identities, for example by Kadyrov and Mashurov and Dougherty-Bliss and Zeilberger.
The comments mention a scathing blog post by Persiflage, condemning the authors' self-promotion and failure to consult experts on continued fractions before claiming that their identities were new. I don't know if the blog post encouraged experts to get involved; possibly it had the opposite effect. At any rate, a number of the formulas have now either been identified as known or been proved, but many seem to be new and remain unproved (e.g., the two examples in the abstract, which have been changed since the original arXiv preprint was posted). IMO it would be good for more number theorists to get involved since the new formulas might be an indication of some new theory that remains to be discovered and elucidated.
As a side remark—a natural question to ask is whether any of these identities might lead to irrationality proofs. The only general theorem in this direction that I know about is a sufficient criterion due to Legendre (see Chrystal's Algebra, page 512, for a proof), and unfortunately none of the Ramanujan Machine's new identities seem to satisfy this criterion. Of course, maybe I overlooked something, or maybe they can be manipulated into a form that does satisfy the criterion.
