On the existence of r-primitive pairs \((\alpha ,f(\alpha ))\) in finite fields

Let r be a divisor of \(q-1.\) An element \(\alpha \in {\mathbb {F}}_{q}\) is said to be r-primitive if ord\((\alpha )=\frac{q-1}{r}\). In this paper, we discuss the existence of r-primitive pairs \((\alpha , f(\alpha ))\) where \(\alpha \in {\mathbb {F}}_q\), f(x) is a general rational function of degree sum m (degree sum is the sum of the degrees of numerator and denominator of f(x)) and the denominator of f(x) is square-free. Then we show that for any integer \(m>0\), there exists a positive constant \(B_{r,m}\) such that if \(q>B_{r,m}\), then such r-primitive pairs exist. In particular, we present a bound for \(B_{r,m}\) with \(r=2\) and \(m\in \{2,3,4,5,6\}\), and provide some conditions on the existence of 2-primitive pairs.