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Simultaneous Primitive Root Values Of Polynomials (2204.02245v1)
Published 4 Apr 2022 in math.GM
Abstract: Let $z\ne \pm1,w2$ be a fixed integer, and let $f(t)\ne g(t)2$ be a fixed polynomial over the integers. It is shown that the subset of primes $p\geq 2$ such that $z$ and $f(z)$ is a pair of simultaneous primitive roots modulo $p$ has nonzero density in the set of primes. The same analysis generalizes to \textit{admissible} $k$-tuple of polynomials $z$, $f_1(z)$, $f_2(z), \ldots$, $f_k(z)$, such that $f_i(z)\ne g_i(z)2$, and $k\ll \log p$ is a small integer.