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.

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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!}) $



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