Asymptotic distance between $x^2+1$ primes? As I recall, the most common difference between consecutive primes starts with $2$, moves on to $6$, then $30$, and is conjectured to progress like the primorials over a long time, without bound.
Is there any analogous prediction about $x^2+1$ or other polynomial prime gaps?
 A: Those things are conjectured by the random model for the primes, the model on which every conjecture about primes is based. This model is saying that when $n$ is picked uniformly in $[1,N]$, for $p\le N^r$ then $n\bmod p$ is uniformly distributed in $1,\ldots, p$ and it is independent from one $p$ to the other. Of course this model is wrong for a fixed $N$, what we assume is that it gets less wrong as $N\to \infty$, ultimately giving correct predictions.


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*What is $r$ ? It is the constant $r\in [1/2,1)$ such that the probability that $n$ (picked uniformly in $[1,N]$) is prime is $$\prod_{p \le N^r} Pr(p\nmid n)=\prod_{p \le N^r} (1-p^{-1})$$
This $r$ can be evaluated from Mertens' theorem and the PNT : it is $r=e^{-\gamma}$, the only constant such that $$\lim_{N \to \infty}\frac{\prod_{p \le N^r} (1-p^{-1})}{\pi(N)/N}=\lim_{N \to \infty}\log N\prod_{p \le N^r} (1-p^{-1})=1$$

*The probability that $n,n+2$ are both primes is
$$\prod_{p \le N^r} Pr(p\nmid n,p\nmid n+2)=(1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})$$
The conjectured number of twin primes $\le N$ is $$C_2\frac{N}{\log^2 N}\sim N(1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})$$ where $C_2$ is the twin prime constant $$C_2=\lim_{N\to \infty}(\log^2 N) (1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})=\lim_{N\to \infty} \frac{(1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})}{\prod_{ p \le N^r} (1- p^{-1})^2}$$

*The probability that $n^2+1$ is prime is $$\prod_{p\le N^{2r}} Pr(p\nmid n^2+1)= (1-2^{-1})\prod_{p\le N^{2r},p\equiv 1\bmod 4} (1-2 p^{-1})$$
Thus the conjectured number of primes $n^2+1$ with $n\le N$ is $$A \frac{N}{\log N}$$ where the obtained constant is $$A = \lim_{N\to \infty}(\log N) (1-2^{-1})\prod_{p\le N^{2r},p\equiv 1\bmod 4} (1-2 p^{-1})$$ $$= \lim_{N\to \infty}\frac{(1-2^{-1})\prod_{p\le N^{2r},p\equiv 1\bmod 4} (1-2 p^{-1})}{\frac12 \prod_{p\le N^{2r}}(1-p^{-1})}$$
And the conjectured mean distance between two consecutive such primes $n^2+1,n_2^2+1$ is $$A\log n$$
