Arithmetic Progressions of r-Primitive Elements in a Field

Article Type

Research Article

Publication Title

Bulletin of the Brazilian Mathematical Society

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 Fq. In fact, we find a condition on q for the existence of α∈Fq× for a given n⩾2 and β∈Fq× such that each of α,α+β,α+2β,⋯,α+(n-1)β⊂Fq× is r-primitive in Fq×. 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∈N, for all but finitely many prime powers q, for any β∈Fq×, there exists α∈Fq such that α,α+β,⋯,α+(n-1)β are all r-primitive whenever r∣q-1. The number of arithmetic progressions in Fq consisting of r-primitive elements of length n, is asymptotic to q(q-1)nφ(q-1r)n.

DOI

10.1007/s00574-024-00412-9

Publication Date

9-1-2024

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