# More Prime Congruences

Find all primes p and q such that

- 12 August, 2012 -
- Article, Math Problems -
- Tags : 104 problems, number theory, titu
- 0 Comments

We first show the trivial property that .

Assume by way of deriving a contradiction that , then it must be the case that . This however violates the assumption that are primes.

Now, like the previous post, we would like to find some congruence class in order to bound and to some class of primes. We would like for the congruence to be derived from the problem statement itself.

The last line of the above derivation comes from the definition of congruence as .

Now, the problem statement is equivalent to

Now, it’s obvious that , therefore, if , and is coprime to , then . This leaves the following possibilities:

Therefore, only satisfies .