Let $(p, p+2)$ denote a pair of twin primes for $p > 3$. Consider a triplet $(2, 5, p)$ and consider a combination in which we take product of any two of these and add third to it. Thus, we get following three combinations: $2p + 5$, $5p+2$ and $p+10$. Out of curiosity, I tried to see the number of primes I get as I vary $p$. If we take any prime $p$ other than the first member of a twin prime pair, then we see that all three, exactly two, exactly one or none could be primes. For example, take $p = 7$ and you would find that all three combinations yield prime numbers.
Surprisingly, when $p$ is the first member of a twin prime pair other than $(3, 5)$, none of these combinations seems to be primes. This is not at all obvious to me. To be precise, I checked this for all twin primes less than a billion ($10^9$). Thus, numerical evidence does suggest that these combinations are always composites. Is there any way to prove this? Thank you
P.S. I can make the codes available to anybody interested, however this is a rather lengthy calculation (on an actual 32 core machine it took 2 hours excluding the time needed to generate these primes!).
Look at the expressions mod $3$. If $p$ and $p+2$ are each prime, then $p$ cannot be $1$ mod $3$, so must be $2$ mod $3$. But then each of the three expressions $2p+5, 5p+2, p+10$ are divisible by $3$.