For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n \geq 5$, we establish a condition sufficient to ensure the existence of a primitive normal pair $(\epsilon,f(\epsilon))$ in $\mathbb{ F}_{ q^{ n}}$ over $\mathbb{ F}_{ q}$ such that $\mathrm{ PN}_{ q^n/q}(\epsilon)=a$, where $a\in\mathbb{ F}_{ q}$ is prescribed. Here $f={ f_{ 1}}/{ f_{ 2}}\in\mathbb{ F}_{ q^n}(x)$ is a rational function subject to some minor restrictions such that deg($f_{ 1}$)+deg($f_{ 2}$)$=m$ and $\mathrm{ PN}_{ q^n/q}(\epsilon) =\sum_{ i=0}^{ n-1}\Bigg(\underset{ j\neq i}{ \underset{ 0\leq j\leq n-1}{ \prod_{ }^{ }}}\epsilon^{ q^j}\Bigg)$. Finally, we conclude that for $m=3$, $n\geq 6$, and $q=7^k$ where $k\in\mathbb{ N}$, such a pair will exist certainly for all $(q,n)$ except possibly $10$ choices at most.
Categories: math.NT