If p is a prime number, and k is an even integer, what is the probability p+k is a prime number?
According to my simulations p+108 is prime twice as often as p+344
If p is a prime number, and k is an even integer, what is the probability p+k is a prime number?
According to my simulations p+108 is prime twice as often as p+344
Note that if $p \equiv 1 \mod 3$, $p + 344$ is divisible by 3 and so must be composite. On the other hand, $p + 108 \equiv p \mod 3$. Thus primes $p$ with $p+344$ prime can occur in only one residue class mod 3, but those with $p+108$ prime can occur in two residue classes mod 3. On the other hand, $p + 344 \equiv p \mod 43$. So I would expect that primes $p + 108$ would occur $(2/1) (41/42) = 41/21$ times as often as $p + 344$.
Technically, the answer to your question is zero. If you fix $k$ and take the number of primes $p$ less than $N$ such that $p+k$ is prime and divide it by the number of primes less than $N$ and take the limit as $N$ goes to infinity, you get zero.