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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?

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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!}) $

$=e$

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