On Diophantine $m$-tuples related to primitive elements of finite fields

Inspired by recent works on Diophantine tuples over finite fields, in this paper we consider Diophantine tuples related to primitive elements of finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q^*=\mathbb{F}_q\setminus\{0\}$ be the multiplicative cyclic group of all non-zero elements over $\mathbb{F}_q$. An element $g\in\mathbb{F}_q$ is called primitive if $g$ generates the group $\mathbb{F}_q^*$. A set $\{x_1,x_2,\cdots,x_m\}\subseteq\mathbb{F}_q^*$ of $m$ elements is said to be a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ if $x_ix_j+1$ is primitive for any $1\le i\le j\le m$. Let $N_m$ denote the number of $\mathcal{P}$-Diophantine tuples over $\mathbb{F}_q$. Then we obtain the asymptotic formula $$m!\cdot N_m=\left(\frac{\varphi(q-1)}{q-1}\right)^{m(m+1)/2}q^m+O_{m,r}\left(q^{m-\frac{1}{2}+r}\right),$$ where $\varphi(\cdot)$ is the Euler totient function and $r\in(0, 1/2)$ is an arbitrary real number. Moreover, we prove that there exists a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ whenever $q\ge \exp(\exp(m(m+1)))$.