Arithmetic Progressions of r-Primitive Elements in a Field

Abstract

In this paper, we deal with the existence of r-primitive elements, a generalisation of primitive elements, in arithmetic progression by using a new formulation of the characteristic function for r-primitive elements in \(\mathbb {F}_q\). In fact, we find a condition on q for the existence of \(\alpha \in \mathbb {F}_q^\times \) for a given \(n\geqslant 2\) and \(\beta \in \mathbb {F}_q^\times \) such that each of \(\alpha , \alpha +\beta ,\alpha +2\beta , \dots , \alpha + (n-1)\beta \subset \mathbb {F}_q^\times \) is r-primitive in \(\mathbb {F}_q^\times .\) This result is utilized with the help of an inequality due to Robin also to produce an explicit bound on q; this, in turn, shows that for any \(n, r\in \mathbb {N},\) for all but finitely many prime powers q, for any \(\beta \in \mathbb {F}_q^\times \), there exists \(\alpha \in \mathbb {F}_q\) such that \(\alpha ,\alpha +\beta ,\dots ,\alpha +(n-1)\beta \) are all r-primitive whenever \(r \mid q-1\). The number of arithmetic progressions in \(\mathbb {F}_q\) consisting of r-primitive elements of length n, is asymptotic to \(\frac{q}{(q-1)^n}\varphi (\frac{q-1}{r})^n\).