This morning I wondered about next question. I am going to ask it as a soft question and as a reference request. My main goal is to know if it is interesting.
One of the Landau's problems is to state if there are infinitely many primes of the form $n^2+1$. See it from this Wikipedia. On the other hand I know other open problem, the so-called Rassias' conjecture, see this Wikipedia.
A well known important heuristic is the so-called abc conjecture. I am saying the version ABC Conjecture II from the Wikipedia's article dedicated to the abc conjecture.
Question. Could be potentially interesting a combination betweeen the statement of the mentioned Landau's problem (the form of such primes $n^2+1$) and Rassias' conjecture? If it was in the literature refer it, and I try to find and read those statements and calculations, in other case provide us your reasonings or heuristic to know if such combination that I've evoked is interesting in the context of primes conjectures. Many thanks.
I do not know which exactly connection do you want to see. A completely elementary and short approach would be:
Suppose that $p=n^2+1$ is prime. Then, if Rassias´ conjecture is true we have:
$p=n^2+1= \dfrac {p_1+p_2+1}{p_1}$. for some primes $p_1<p_2$
That is, we have $n^2p_1-p_2=1$.
Because $\gcd (n^2,-1)=1$ Bezout´s identity guarantees that we have an infinite number of solutions $(x,y)$ to the equation $n^2x-y=1$ but here, we need both of them to be positive different primes.