# Chen primes and permutations

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.