# Thread: Pairs of primes and Cardinalities of sets

1. ## Pairs of primes and Cardinalities of sets

For a pair of primes $(p_0,p_1)$, let Condition $A$ be the following:
For all integers $n>0$, $\mbox{card}\left( \left\{ p_{1-j}^i \in \mathbb{Z} / p_j^n \mathbb{Z} \mid i \in \mathbb{N} \right\} \right) = p_j^n-p_j^{n-1}$ for both $j=0$ and $j=1$.

Let $M$ be the set of all pairs of primes that satisfy Condition $A$. I can show that $(2,3) \in M$, so $M$ is nonempty. I think it might be true that $(p_0,p_1) \in M$ if $p_0 and there does not exist a prime $p_2$ such that $p_0.

Let $K = \left\{(p_0,p_1) \mid p_0,p_1 \mbox{ prime}, p_0. Then I think it may be true that $M = \left\{(p_0,p_1) \mid (p_0,p_1) \in K \mbox{ or } (p_1,p_0) \in K\right\}$.

Is there an obvious counterexample? (So far, I haven't found any, but I admit I haven't looked terribly hard yet).

2. ## Re: Pairs of primes and Cardinalities of sets

Ok, I figured this out. I am essentially looking for pairs of primes $p,q$ for which $p$ is a primitive root modulo $q^n$ for all $n>0$ and $q$ is a primitive root modulo $p^n$ for all $n>0$. I realized that $(2,3)\notin M$, I have no clue what I was thinking. And it is possible that $M$ is empty. My bad. Thanks anyway.