# Math Help - Prime factors

1. ## Prime factors

Hello!

I've got a little problem with the following assignment. I would be very grateful if anyone could at least give me some hint on how to do it.

There are given different odd prime numbers $p$ and $q$. Prove that number $2 ^ {pq} - 1$ has at least 3 different prime factors.

Thank you very much!

2. Originally Posted by PaulinaAnna
Hello!

I've got a little problem with the following assignment. I would be very grateful if anyone could at least give me some hint on how to do it.

There are given different odd prime numbers $p$ and $q$. Prove that number $2 ^ {pq} - 1$ has at least 3 different prime factors.

Thank you very much!
Maybe this?

Mersenne prime - Wikipedia, the free encyclopedia (subheading Generalization)

By the way I've marked this thread to be moved to the Number Theory subforum.

3. Let $m$ and $n$ be positive integers, if $m|n$, then $2^m-1|2^n-1$.
Also, for any positive integers $m$ and $n$, if $(m,n)=1$, then $(2^n-1,2^m-1)=1$.

Apply these two lemmas in the same order they're written...

4. ## Already solved

Hey!

Thank you guys very much.

It's already solved

5. ## Question

Hey, I'm wondering how did you solve this problem? I'm interested, I tried with Mersenne numbers, but I didn't manage to prove that this number has at least 3 prime divisors.

6. Anyway, I got it. It was quite easy.

7. Err... No. My question remains- I "solved" it in the wrong way, now I realised. How to prove this fact?