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$\begingroup$@FedorPetrov actually it only asks for one prime $p$ so that $p+n^2$ is prime, not infinitely many (which is the case for the twin primes conjecture/bounded gaps theorem). So the $n=2$ case is very much solved!$\endgroup$
$\begingroup$in the spirit of the question, I asked chatGPT 4 if this conjecture was known to be true/false or open. It answered: "Up until my last training data in 2021, this conjecture was not resolved. For the most current and accurate information, you should refer to recent mathematical literature or ask a professional mathematician or number theorist."$\endgroup$
$\begingroup$@BoazTsaban to me, this is like saying "I rolled some dice and I used the results as coefficients for some polynomial, what is the Galois group of this polynomial". LLMs by their nature cannot understand their output, so saying this is a "conjecture by ChatGPT" seems to be attributing agency/sentience where it doesn't exist.$\endgroup$
$\begingroup$@BoazTsaban chatGPT is nothing else but a text generator generating the most probable sequel of a sequence of words. You can play with the openAI playground to see that. Hence all hallucinations that this so-called AI has: it definitively cannot be trusted. When the sequence of words looks like a question, the most probable sequel is a sequence of words which looks for us, humans, like an answer because of its training. chatGPT does not understand what it's talking about.$\endgroup$
Ancient Greeks conjectured that there are infinitely many pairs of primes which differ by 2 (twin primes). A natural widely believed generalization is that 2 may be replaced by every even number. Moreover, it is expected that for every integer $T>0$ there are about $C n/\log^2 n$ numbers $p\leqslant n$ for which $p$ and $p+2T$ are both primes.
What is proved is that there are infinitely many pairs of primes with the same difference (Zhang and beyond). But it is not known for difference 2 or 4. If you (or chatgpt) need only one pair, this is certainly checked for all not too large integers, but I am afraid that still open for large enough integers.
$\begingroup$@GeraldEdgar well, I was sure in this as a common knowledge, but now I am less certain. Greeks are mentioned in several places (see below, for example), but without citation and assurance. nature.com/articles/nature.2013.12989$\endgroup$
$\begingroup$About twin primes, the Nature snippet says, in passing, "Some attribute the conjecture to the Greek mathematician Euclid of Alexandria". That seems not convincing. Euclid did not even say "there are infinitely many primes"; he said "Given a [finite] list of primes, to construct a prime not in the list."$\endgroup$
The more natural conjecture, that every even number arises as the difference of two primes, was asked on math.SE years ago. That question remains open, and I have a hard time imagining that restricting to squares makes the problem any more tractable.
EDIT: The conjecture that every even number is the difference of two primes is apparently sometimes called Maillet's conjecture.
$\begingroup$I believe this conjecture predates the coming of stackexchange by many decades. Already in 1849 de Polignac conjectured that every even number arises infinitely often as the difference of consecutive primes.$\endgroup$
$\begingroup$@GerryMyerson Good point. On the other hand, conceivably the question of whether every even number arises at least once as the difference of two not-necessarily-consecutive primes could be easier than Polignac's conjecture.$\endgroup$
