Pairs of primes and Cardinalities of sets
For a pair of primes , let Condition be the following:
For all integers , for both and .
Let be the set of all pairs of primes that satisfy Condition . I can show that , so is nonempty. I think it might be true that if and there does not exist a prime such that .
Let . Then I think it may be true that .
Is there an obvious counterexample? (So far, I haven't found any, but I admit I haven't looked terribly hard yet).
Re: Pairs of primes and Cardinalities of sets
Ok, I figured this out. I am essentially looking for pairs of primes for which is a primitive root modulo for all and is a primitive root modulo for all . I realized that , I have no clue what I was thinking. And it is possible that is empty. My bad. Thanks anyway.