Probability p+k is a prime
Solution 1:
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 $p + 344$.