The unproved formulas of Ramanujan Are there any formulas due to Ramanujan that have still not been proved—or disproved?
If so, what are they?
I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^x$ and $3^x$ are both integers then $x$ is an integer.  There may be other open conjectures due Ramanujan.  However, right now I'm mainly interested in formulas, i.e. identities, that he wrote down.
 A: As far as I know, at least the following Ramanujan's claim about mock theta functions has not been proved, which appeared in a letter from Ramanujan to Hardy. Ramanujan claimed that: Let $q=e^{-t}$, then one has an asymptotic expansion of form
\begin{align}1+\frac{q}{(1-q)^2}+&\frac{q^{3}}{(1-q)^2(1-q^2)^2}+\frac{q^{6}}{(1-q)^2(1-q^2)^2(1-q^3)^2}+\cdots\\
&=\sqrt{\frac{t}{2\pi\sqrt{5}}}\exp\left(\frac{\pi^2}{5t}+\frac{t}{8\sqrt{5}}+a_2t^2+a_3t^3+\cdots+O(a_kt^k)\right),\; t\rightarrow 0^+,
\end{align}
with infinity many $a_k\neq 0$. See pages 57-58 of [Watson, G. N. The Final Problem : An Account of the Mock Theta Functions. J. London Math. Soc. 11 (1936), no. 1, 55–80.] for details.
A: I don't have enough reputation to comment. Has this  expression for $\sqrt{\pi e^x/2x}$ been proved?
A: Manjul Bhargava says only half done around 54:39 of the video Manjul Bhargava, Steven Strogatz, Matt Brown and Lynn Sherr — The Infinite Mind from March 2016.
A: George Andrews and Bruce Berndt have written five books about  Ramanujan's lost notebook, which was actually not a notebook but a pile of notes Andrews found in 1976 in a box at the Wren Library at Trinity College, Cambridge.   In 2019 Berndt wrote about the last unproved identity in the lost notebook:

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*Bruce C. Berndt, Junxian Li and Alexandru Zaharescu, The final problem: an identity from Ramanujan's lost notebook, Journal of the London Mathematical Society 100 (2) (2019), 568–591.

Following Timothy Chow's advice, I consulted Berndt and asked him if there were any remaining formulas of Ramanujan that have neither been proved nor disproved.  He said no:

To the best of my knowledge, there are no claims or conjectures remaining.  There are some statements to which we have not been able to attach meaning.

I checked to make sure that this applies to all of Ramanujan's output, not just the lost notebook, and he said yes.
EDIT: However, only on December 21st, 2021 did Örs Rebák submit this paper to the arXiv:

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*Örs Rebák, The three missing terms in Ramanujan's septic theta function identity.

in which he completed an incomplete formula in Ramanujan's lost notebook, and proved it.  So there may still be gems left to polish.
A: Bruce Berndt has claimed that all the claims in Ramanujan's "Lost Notebook" have been proved, with a solution to the the final problem being published by Berndt, Li, and Zaharescu in J. London Math. Soc. in 2019. However, I am not sure that this means that all the formulas in Ramanujan's other writings have been proved. If you have not yet tried directly writing to Bruce Berndt, that would be my first suggestion.
