, 2021

First, construct distinct positive rational numbers $r_1, r_2, \dots, r_{k+3}$ such that $r_i r_{i+1} r_{i+2} r_{i+3} = i$ for all $i = 1, 2, \dots, k$. Since $r_{i+4} = \frac{i+1}{i} r_i$, $\{r_{4n+r}\}_n$ are strictly increasing sequences for $r = 1, 2, 3, 4$. Choose $r_1, r_2, r_3$ to be distinct primes larger than $k$ and $r_4 = \frac{1}{r_1 r_2 r_3}$, it is easy to see that $\{r_i\}$ are distinct rational numbers. Write $r_i = \frac{u_i}{v_i}$ as irreducible fractions. It is easy to see that $u_i (v_i)^{-1} = u_j (v_j)^{-1}$ in $\mathbb{Z}_p$ only if $u_i v_j \equiv u_j v_i \pmod{p}$. Choose prime $p > (\max_i\{u_i, v_i\})^2$, then $a_i = u_i (v_i)^{-1}$ in $\mathbb{Z}_p$ satisfies the desired properties.