TITLE: The Challenge of the Twin Prime Conjecture
QUESTION [5 upvotes]: Are there an infinite amount of twin composite pairs (which are infinite)  (https://oeis.org/A060461) whose prime factors are contained within a twin prime pair?
Example: $ (119,121) $
$ 119 = 7 * 17  $ 
$ 121 = 11 * 11 $
$7, 11, 17$ are contained within $(5,7)$ $(11, 13)$ and $(17, 19)$ which are twin  prime pairs
Are questions like this impossible to use to answer the Twin Prime Conjecture?

REPLY [5 votes]: Probably yes, on probabilistic grounds; and certainly yes on the Hardy–Littlewood prime k-tuples conjecture, which implies (for example) that there are infinitely many integers $n$ for which $3n-1$, $3n+1$, $5n-1$, and $5n+1$ are all prime. In that case, $3(5n+1)$ and $5(3n+1)$ are integers that differ by two, all of whose prime factors are part of twin prime pairs.
However, if there are only finitely many twin primes, then it is probably possible to prove that the answer is "no" (this sounds like a job for "S-unit equations"). Assuming this is correct, then the problem in the OP is harder than the twin primes conjecture.