Are there infinitely many Thâbit ibn Kurrah cousin primes?

Question

Positive integers of the form $3 * 2^n - 1$ are called Thâbit ibn Kurrah numbers.

and if those numbers are prime they are called Thâbit ibn Kurrah primes.

Now if for a fixed positive integer $n$ , $3 * 2^n - 1$ and $3 * 2^n - 5$ are both prime then we have a Thâbit ibn Kurrah cousin prime.

Conjecture : there are infinitely many Thâbit ibn Kurrah cousin primes.

$7,11$

$19,23$

$41,47$

$379,383$

Are there more known ??

Although its not reasonable to expect a proof for the conjecture since prime twins and cousin primes are not proven to be infinite sequences , it might be possible to disproof the conjecture.

But I was not able to find a reason why the conjecture would fail.

I know that there are only a finite amount of prime twins of type $2^n + 3$,$2^n + 5$ because from working with mod $24$ you can show that at least one of the two must be divisible by $3,5,7,11$ or $13$. ( and actually $11$ can be dropped ).

But I was not able to find such an argument for the Thabit ibn Kurrah cousins.

Answer 1

The following is a heuristic based on the Borel-Cantelli lemma that suggests that there will only be finitely many such primes. I believe that actually proving such a result would be very difficult.

Assuming independence, the probability that both $3*2^n-1$ and $3*2^n-5$ are prime is $$\approx \frac{C_1}{(\log(2^n))^2}=\frac{C_2}{n^2}$$ for some constants $C_1,C_2$. Now, the sum $$\sum_{n=1}^\infty \text{Pr}(E_n)=\sum_{n=1}^\infty \frac{C_2}{n^2}$$ converges, and so by the Borel-Cantelli lemma we expect that there are only finitely many such cousins.