Simultaneous Primitive Root Values Of Polynomials

2022/04/04 by Carella, N. A.
#11N13 #11N37 #FOS: Mathematics #General Mathematics (math.GM) #Primary 11A07 #Secondary 11N05

paper · doi:10.48550/arxiv.2204.02245

Abstract

Let z≠ ±1,w² be a fixed integer, and let f(t)≠ g(t)² be a fixed polynomial over the integers. It is shown that the subset of primes p≥ 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 admissible k-tuple of polynomials z, f₁(z), f₂(z), …, f_k(z), such that f_i(z)≠ g_i(z)², and k≪ log p is a small integer.

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