Abstract

Let be a finite field with qn elements. For a positive divisor r of , the element is called r-primitive if its multiplicative order is . Also, for a non-negative integer k, the element is k-normal over if in has degree k. In this paper we discuss the existence of elements in arithmetic progressions with being ri -primitive and at least one of the elements in the arithmetic progression being k-normal over . We obtain asymptotic results for general and concrete results when for .

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