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.
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.