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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?

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    $\begingroup$ Those pairs $(6n-1,6n+1)$ both composite. Why the prime factors would be related to the twin primes ? The idea is if we can estimate the density of primes and of twin composites, we can deduce the density of twin primes. $\endgroup$
    – reuns
    Jul 30, 2017 at 23:42
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    $\begingroup$ Heuristically, the set of primes that belong to prime pairs might be $A \cup B$ where $A, B$ are finite, and yet there might be infinitely many composite $n$ such that all the prime factors of $n$ belong to $A$ and all the prime factors of $n+2$ belong to $B.$ $\endgroup$
    – DanielWainfleet
    Jul 30, 2017 at 23:55
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    $\begingroup$ Interesting: $$|5^2-3^3|=|7^2-3\cdot17|=|11^2-7\cdot17|=|13^2-3^2\cdot19|=|17^2-7\cdot41|=|19^2-3\cdot11^2|=2$$ (I haven't checked any further than that.) $\endgroup$
    – Barry Cipra
    Jul 31, 2017 at 1:22
  • $\begingroup$ Not necessarily, it is very slightly possible that you could first find a proof that there are infinitely many pairs of odd composite numbers 2 apart where all of its prime factors are part of a twin prime pair, and then from that, prove the twin prime conjecture. $\endgroup$
    – Timothy
    Apr 6, 2020 at 0:56

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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.

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    $\begingroup$ Suppose there exist non-empty finite $S=\{s_1,...,s_m\}$ with $m$ members and $T=\{t_1,...,t_n\}$ with $n$ members, where each member of $S\cup T$ belongs to a prime pair, such that there are infinitely many $(d_1,..,d_m), (e_1,...,e_n)$ with each $d_i$ and $e_j$ in $\{0\}\cup \Bbb N,$ satisfying $\prod_{i=1}^ms_i^{d_i} -\prod_{j=1}^n t_j^{e_j}=\pm 2.$ This would give a "Yes" to the first sentence of the Q without settling the Twin Prime Conjecture. $\endgroup$
    – DanielWainfleet
    Feb 2, 2018 at 19:02

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