Prime factors

**PaulinaAnna** · September 6th 2010, 09:48 AM

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!

**undefined** · September 6th 2010, 11:55 AM

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.

**melese** · September 6th 2010, 02:53 PM

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...

**PaulinaAnna** · September 7th 2010, 05:28 AM

Hey!

Thank you guys very much.

It's already solved

**conrad** · November 19th 2010, 10:44 AM

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.

**conrad** · November 19th 2010, 01:19 PM

Anyway, I got it. It was quite easy.

**conrad** · November 20th 2010, 04:57 AM

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

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