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In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes.

For $k=1,2,3,\ldots$ let $p_k$ denote the $k$-th prime. Motivated by Question 315581 of mine (cf. Permutations $\pi\in S_n$ with $p_k+p_{\pi(k)}+1$ prime for all $k=1,\ldots,n$), here I raise a question involving Chen primes and permutations.

QUESTION: Is my following conjecture true?

Conjecture. (i) For any positive integer $n$, there is an even permutation $\sigma$ of $\{1,\ldots,n\}$ such that $p_kp_{\sigma(k)}-2$ is prime for every $k=1,\ldots,n$.

(ii) For any integer $n>2$, there is an odd permutation $\sigma$ of $\{1,\ldots,n\}$ such that $p_kp_{\sigma(k)}-2$ is prime for every $k=1,\ldots,n$.

Let $a_0(n)$ (resp., $a_1(n)$) denote the number of even (resp., odd) permutations $\sigma$ of $\{1,\ldots,n\}$ with $p_kp_{\sigma(k)}-2$ prime for all $k=1,\ldots,n$. Via Mathematica I find that $$(a_0(1),\ldots,a_0(11))=(1,1,1,1,3,6,1,1,33,125,226)$$ and $$(a_1(1),\ldots,a_1(11))=(0,0,1,2,2,6,1,2,32,123,222).$$ Also, the values of $a_0(n)+a_1(n)$ for $n=12,\ldots,27$ are \begin{gather}1792,\ 4288,\ 6468,\ 27068,\ 29752,\ 106066,\ 447982, \\1250762,\ 6304196,\ 46613084,\ 126391780,\ 504582496 \\2270372946,\ 3028652541,\ 8941959118,\ 36442298864 \end{gather} respectively. For example, when $n=7$, the only even permutation of $\{1,\ldots,7\}$ meeting the requirement is $(1,5,7,4,2,6,3)$ with \begin{gather}p_1p_1-2=2,\ p_2p_5-2=31,\ p_3p_7-2=83, \\ p_4p_4-2=47,\ p_5p_2-2=31,\ p_6p_6-2=167,\ p_7p_3-2=83\end{gather} all prime, and the the only odd permutation of $\{1,\ldots,7\}$ meeting the requirement is $(1,5,7,6,2,4,3)$ with \begin{gather}p_1p_1-2=2,\ p_2p_5-2=31,\ p_3p_7-2=83, \\ p_4p_6-2=89,\ p_5p_2-2=31,\ p_6p_4-2=89,\ p_7p_3-2=83\end{gather} all prime.

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