I asked Chat GPT to suggest a number theoretic conjecture.
It came up with the following interesting conjecture:


**Conjecture (Chat GPT):** For each even natural number $n$, there is a prime $p$ such that $p+n^2$ is also prime.


(I am not sure it stated *even* but this is clearly necessary, e.g., take $n=5$.)


Equivalently, if $P$ is the set of primes, then the difference set $P-P$ contains all squares of even natural numbers.


Is this conjecture true or false?


What is known about variations of this conjecture, with some other function $f(n)$ instead of $n$?


I am not a number theorist, but every mathematician finds number theoretic questions interesting.



A number theoretic conjecture by Chat GPT
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.


