Di Han and Wenpeng Zhang: On the existence of some special primitive roots mod p, p.59-66

Abstract:

The main purpose of this paper is using the properties of Gauss sums and the estimate for character sums to study the properties of the primitive roots of $p$ (an odd prime), and prove that for any integers $k \neq 1$ and $(mn, p)=1$, there exists a primitive root $\xi$ of $p$ such that $m\xi^k+n\xi$ is also a primitive root of $p$, provide $p$ large enough. Let $N(k, m, n; p)$ denotes the number of all primitive roots $\xi$ of $p$ such that $m\xi^k+n\xi$ is also a primitive root of $p$. Then we can also give an interesting asymptotic formula for $N(k, m, n; p)$.

Key Words: Primitive root of $p$, Gauss sums, character sums, Asymptotic formula.

2010 Mathematics Subject Classification: Primary: 11M20;
Secondary: 11L40.

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