Recently, Paul Bradley proved in arXiv:1809.01012 that for any positive integer $n$ there is a permutation $\pi_n$ of $\{1,\ldots,n\}$ such that $k+\pi_n(k)$ is prime for every $k=1,\ldots,n$ (cf. http://arxiv.org/abs/1809.01012). Motivated by this, here I pose the following question.

QUESTION: Is my following conjecture true?

**Conjecture**. (i) For any positive integer $n$, there is a permutation $\sigma_n$ of $\{1,\ldots,n\}$ such that $k\sigma_n(k)+1$ is prime for every $k=1,\ldots,n$.

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

I have checked the conjecture for $n$ up to $11$. For example, $(1, 3, 2, 9, 6, 5, 10, 11, 4, 7, 8)$ is a permutation of $\{1,\ldots,11\}$ with \begin{gather}1\times1+1,\ 3\times2+1,\ 2\times3+1,\ 9\times4+1,\ 6\times5+1, \ 5\times 6+1, \\10\times 7+1,\ 11\times8+1,\ 4\times9+1,\ 7\times10+1,\ 8\times11+1 \end{gather} all prime, and $(3, 2, 1, 5, 4, 7, 6, 9, 8, 11, 10)$ is a permutation of $\{1,\ldots,11\}$ with \begin{gather}3\times1-1,\ 2\times2-1,\ 1\times3-1,\ 5\times4-1,\ 4\times5-1, \ 7\times 6-1, \\6\times 7-1,\ 9\times8-1,\ 8\times9-1,\ 11\times10-1,\ 10\times11-1 \end{gather} all prime.

**Remark**. I also conjecture that for any integer $n>2$ there is a permutation $\pi_n$ of $\{1,\ldots,n\}$ such that the $2n$ numbers $k+\pi_n(k)\pm1\ (k=1,\ldots,n)$ are all prime. This is stronger than the twin prime conjecture.