Suppose $d\_1, d\_2$ are two fixed coprime integers, $\frac{d\_1}{d\_2} \neq \pm 1$. Given any $n > 0$, can we find a prime number $p$ such that the order of $d\_1d^{-1}\_2$ in the multiplicative group of the field $\mathbb{Z}/p\mathbb{Z}$ be greater than $n$?



Can the order of a rational number in Z/pZ be as large as we want
The answer to your question is "yes" (cf. Douglas Zare's comment). In fact, for all sufficiently large primes $p$, the order of $d\_1 d\_2^{-1}$ is greater than $n$. Here, "sufficiently large" means greater than $|d\_1|^n$ and $|d\_2|^n$.


