We are given the twin primes $a$ and $b$, where $a > 5$. We are told that only one of the following: $ab-3, ab-2, ab-1, ab+1, ab+2, ab+3$ will sometimes generate a prime but not always.
It's clear straight away that ab-3, ab+3, ab-1 and ab+1 can never be prime since ab when multiplied together is odd and therefore adding an odd to odd gives you an even number so therefore never prime.
That leaves only $ab-2$ and $ab+2$ that could possibly generate a prime number. Through some calculations I found it can only be $ab+2$ but I can't seem to find a proof of why, I'm guessing it's something to do with the fact that $ab-2$ but be a multiple of 3 or 5 but I'm not 100% sure.
Any help would be greatly appreciated and if anything is unclear please let me know, cheers.