If $n$ is a prime let's define (we are in radians):
$$\lfloor n\sin^{2}\left(n\right)+1\rfloor$$
Do we have infinitly primes of the form $$\lfloor n\sin^{2}\left(n\right)+1\rfloor$$
For small primes we have :
$(11,11),(13,3),(61,57),...$
As a random trial, we have an example: $n=100043$. It gives us $72467$, which is a prime. In this case we have: $$100043=2+3 \cdot 33347$$ $33347$ is a prime, so we have a new pair: $(33347,17837)$.
Perhaps we can use a famous result due to Euler:
$$x\sin^{2}\left(x\right)=x\left(x\prod_{n=1}^{\infty}\left(1-\frac{x^{2}}{n^{2}\pi^{2}}\right)\right)^{2}$$
Following the advice @Peter if $1<y<2$:
$$y=x\sin^{2}\left(x\right)+1=\exp\left(-\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}\left(x\sin^{2}\left(x\right)\right)^{n}}{n}\right)$$
If We have :
$$f_{mi}\left(x\right)=\exp\left(-\sum_{n=1}^{i}\frac{\left(-1\right)^{n}\left(x\left(x\prod_{k=1}^{m}\left(1-\frac{x^{2}}{k^{2}\pi^{2}}\right)\right)^{2}\right)^{n}}{n}\right)$$
Then $C$ a positive integer :
$$f_{mi}(C\pi)=f_m(C\pi)f_i(C\pi)=1$$
Perhaps it's a multiplicative function https://en.wikipedia.org/wiki/Multiplicative_function .But I'm not sure...
Nota bene :
For $p$ an integer sufficiently large ,$i,m\in[p,\infty]$ $\exists \varepsilon \in[0,1],x\in[C\pi-\varepsilon ,C\pi+\varepsilon]$ such that :
$$x\sin^2(x)+1\leq \exp\left(-\sum_{n=1}^{i}\frac{\left(-1\right)^{n}\left(x\left(x\prod_{k=1}^{m}\left(1-\frac{x^{2}}{k^{2}\pi^{2}}\right)\right)^{2}\right)^{n}}{n}\right)$$
Last attempt :
We need to find see Willans formula https://en.wikipedia.org/wiki/Formula_for_primes :
$$p_n-1=\sum_{i=1}^{2^{n}}\operatorname{floor}\left(\left(\frac{n}{\sum_{j=1}^{i}\operatorname{floor}\left(\left(\cos\left(\frac{\left(\left(j-1\right)!+1\right)}{j}\pi\right)\right)^{2}\right)}\right)^{\frac{1}{n}}\right)=\lfloor k\sin^2(k)\rfloor$$
Do we have infinite primes of this kind ? If yes, how do we show it? As in my trial example, what is the maximum number of pairs we can make if we subtract by two and divide by three?
PS: Can you propose a name for this kind of pair of primes?