In this paper, we explore the existence of m-term arithmetic progressions in \({\mathbb {F}}_{q^n}\) with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for \(m \ge 4\) and concrete results for \(m \in \{2,3\}\), where the complete list of exceptions when the common difference belongs to \({\mathbb {F}}_{q}\) is obtained. The proofs combine character sums, sieve estimates and computational arguments using the software SageMath.

中文翻译：

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关于有限域中的算术级数

在本文中，我们探索了\({\mathbb {F}}_{q^n}\)中具有给定公差的m项算术级数的存在性，其项都是本原元素，并且至少有一个是正常的。我们获得了\(m \ge 4\)的渐近结果和\(m \in \{2,3\}\)的具体结果，其中公差属于\({\mathbb { F}}_{q}\)得到。这些证明使用软件 SageMath 结合了字符总和、筛分估计和计算参数。