Can the order of a rational number in Z/pZ be as large as we want - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:40:50Z http://mathoverflow.net/feeds/question/108683 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108683/can-the-order-of-a-rational-number-in-z-pz-be-as-large-as-we-want Can the order of a rational number in Z/pZ be as large as we want Xiaolei Wu 2012-10-03T02:40:41Z 2012-10-03T09:47:26Z <p>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 $n$?</p> http://mathoverflow.net/questions/108683/can-the-order-of-a-rational-number-in-z-pz-be-as-large-as-we-want/108685#108685 Answer by S. Carnahan for Can the order of a rational number in Z/pZ be as large as we want S. Carnahan 2012-10-03T03:19:24Z 2012-10-03T03:19:24Z <p>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 $|d_2|^n$.</p>