@MatthewBolan Thank you very much! - Wouldn't you like to turn this into an answer? Besides, I also suspected the third question may be the hardest.

@mathworker21 For $x = \sqrt{2}$, one finds the solutions $(p,q) = (3,2)$, $(p,q) = (7,5)$, $(p,q) = (41,29)$, $(p,q) = (63018038201,44560482149)$ and $(p,q) = (19175002942688032928599,13558774610046711780701)$, and there is a chance that these are all (if there are more, $p$ and $q$ have more than $1000$ decimal digits).

Numerical experimentation suggests that e.g. for the square root of 2, there is only a small finite number of pairs of primes which work, which is supported by heuristics.

Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on MathOverflow Meta, or in MathOverflow Chat. Comments continuing discussion may be removed.

Would it be possible to expand this a bit, to make it more self-contained? Link-only answers tend to get completely useless once the link gets broken.

Please be aware that every edit of a question or of one of its answers bumps the thread to the front page. This has happened for this thread already about 20 times in just a few days, and this is a nuisance for other users. Please refrain from unnecessary edits to your posts.

@NateRiver This has already been suggested many times, and has equally many times been declined.

Please be aware that every edit of a question or of one of its answers bumps the thread to the front page. This has happened for this thread already more than 20 times in just a few days, and this is a nuisance for other users. Please refrain from unnecessary edits to your posts.

@Al-Amrani As to account merge, the proper procedure is to contact the CMs via the contact form.