# Have new conjectures generated by the Ramanujan machine been proven?

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?

$$3+\displaystyle \frac{-1}{ \displaystyle 4+ \frac{-2}{ \displaystyle 5+ \frac{-3}{ \displaystyle 6+ \frac{-4}{ 7+ \cdots}}}}$$ $$=\lim_{n\rightarrow\infty}\lim_{x\rightarrow\infty}3+\displaystyle \frac{-1}{ \displaystyle 4+ \frac{-2}{ \displaystyle 5+ \frac{-3}{ \displaystyle 6+ \frac{-4}{ \cdots (n+2)-\frac{n}{x}}}}}$$ $$\displaystyle=\lim_{n\rightarrow\infty}\lim_{x\rightarrow\infty}(\frac{1}{n!}\frac{x-1}{nx+1-n}+\sum_{k=0}^{n}\frac{1}{k!})$$
$$=e$$