Simultaneous Elements Of Prescribed Multiplicative Orders

Abstract

Let $u\ne \pm 1$, and $v\ne \pm 1$ be a pair of fixed relatively prime squarefree integers, and let $d\ge 1$, and $e\ge 1$ be a pair of fixed integers. It is shown that there are infinitely many primes $p\ge 2$ such that $u$ and $v$ have simultaneous prescribed multiplicative orders ${\text{ord}}_{p}u=(p-1)/d$ and ${\text{ord}}_{p}v=(p-1)/e$ respectively, unconditionally. In particular, a squarefree odd integer $u>2$ and $v=2$ are simultaneous primitive roots and quadratic residues (or quadratic nonresidues) modulo $p$ for infinitely many primes $p$, unconditionally.

Twenty Pages. Keywords: Distribution of primes; Primes in arithmetic progressions; Simultaneous primitive roots; Simultaneous prescribed orders; Schinzel-Wojcik problem

References in corpus (3)

• New estimates of double trigonometric sums with exponential functions
• Open Diophantine Problems
• Variations on a theme of Schinzel and Wójcik