Arithmetic Progressions of r-Primitive Elements in a Field

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发表日期：

2024年8月7日

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Jyotsna Sharma,
Ritumoni Sarma,
Shanta Laishram

摘要

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 $\mathbb {F}_q$ $F_{q}$ . In fact, we find a condition on q for the existence of $\alpha \in \mathbb {F}_q^\times $ $α \in F_{q}^{\times}$ for a given $n\geqslant 2$ $n ⩾ 2$ and $\beta \in \mathbb {F}_q^\times $ $β \in F_{q}^{\times}$ such that each of $\alpha , \alpha +\beta ,\alpha +2\beta , \dots , \alpha + (n-1)\beta \subset \mathbb {F}_q^\times $ $α, α + β, α + 2 β, \dots, α + (n - 1) β \subset F_{q}^{\times}$ is r-primitive in $\mathbb {F}_q^\times .$ $F_{q}^{\times} .$ 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 $n, r\in \mathbb {N},$ $n, r \in N,$ for all but finitely many prime powers q, for any $\beta \in \mathbb {F}_q^\times $ $β \in F_{q}^{\times}$ , there exists $\alpha \in \mathbb {F}_q$ $α \in F_{q}$ such that $\alpha ,\alpha +\beta ,\dots ,\alpha +(n-1)\beta $ $α, α + β, \dots, α + (n - 1) β$ are all r-primitive whenever $r \mid q-1$ $r ∣ q - 1$ . The number of arithmetic progressions in $\mathbb {F}_q$ $F_{q}$ consisting of r-primitive elements of length n, is asymptotic to $\frac{q}{(q-1)^n}\varphi (\frac{q-1}{r})^n$ $\frac{q}{{(q - 1)}^{n}} φ {(\frac{q - 1}{r})}^{n}$ .