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.)

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.

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.