# The Challenge of the Twin Prime Conjecture

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?

• 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. Jul 30, 2017 at 23:42
• 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 $B.$ Jul 30, 2017 at 23:55
• 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.) Jul 31, 2017 at 1:22
• 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. Apr 6, 2020 at 0:56

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.
• 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. Feb 2, 2018 at 19:02