On the existence of r-primitive pairs (\alpha ,f(\alpha ))$$ in finite fields (original) (raw)

Abstract

Let r be a divisor of \(q-1.\) An element \(\alpha \in {\mathbb {F}}_{q}\) is said to be _r_-primitive if ord\((\alpha )=\frac{q-1}{r}\). In this paper, we discuss the existence of _r_-primitive pairs \((\alpha , f(\alpha ))\) where \(\alpha \in {\mathbb {F}}_q\), f(x) is a general rational function of degree sum m (degree sum is the sum of the degrees of numerator and denominator of f(x)) and the denominator of f(x) is square-free. Then we show that for any integer \(m>0\), there exists a positive constant \(B_{r,m}\) such that if \(q>B_{r,m}\), then such _r_-primitive pairs exist. In particular, we present a bound for \(B_{r,m}\) with \(r=2\) and \(m\in \{2,3,4,5,6\}\), and provide some conditions on the existence of 2-primitive pairs.

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Acknowledgements

This research is supported by the National Natural Science Foundation of China under Grant Nos. 11771007 and 12171241.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
    Hanglong Zhang & Xiwang Cao
  2. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China
    Xiwang Cao

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  1. Hanglong Zhang
  2. Xiwang Cao

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Correspondence toXiwang Cao.

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Zhang, H., Cao, X. On the existence of _r_-primitive pairs \((\alpha ,f(\alpha ))\) in finite fields.AAECC 35, 725–738 (2024). https://doi.org/10.1007/s00200-022-00585-0

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