Simultaneous Elements Of Prescribed Multiplicative Orders

Let $u\mathrm{\ne }\pm 1$, and $v\mathrm{\ne }\pm 1$ 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 ${\text{ord}}_{p}u=(p-1)\mathrm{/}d$ and ${\text{ord}}_{p}v=(p-1)\mathrm{/}e$ 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.