Two variants of a theorem of Schinzel and Wójcik on multiplicative orders

Schinzel and Wójcik have shown that if $\alpha ,\beta \in \mathbb Q^{\times} \setminus \{\pm 1\}$, then there are infinitely many primes $p$ where $v_p(\alpha )=v_p(\beta )=0$ and where $\alpha ,\beta $ share the same multiplicative order modulo $p$. We present two variants of their result. First, we give a short and simple proof of the analogous statement where $\mathbb Q$ is replaced by any global function field $K$. Second, we show that a similar conclusion holds in the number field case provided one can find a suitable ‘auxiliary prime’. Given $K$, $\alpha $, and $\beta $, it appears simple in practice to find such a prime. As an application, we prove there are infinitely many primes $p$ with the same rank of appearance in the sequences of Pell and Fibonacci numbers.