1. ## Mersenne numbers

Hello. I've got three questions.
1.Can a Mersenne number be a multiple of another Mersenne number? Prove. (I mean is that possible that 2^a - 1 | 2^b - 1 where a, b are prime numbers)
2.Can a Mersenne number be equal to X^n ? (X and n are integer). Prove. (This Mersenne number is 2^d - 1 where d is a prime number)
3.Is it possible that (2^t - 1)^a * (2^u - 1)^b is equal to 2^tu - 1? (t, u, 2^t -1, 2^u -1 are prime numbers).

2. 1. If $\displaystyle p_{1}[/Math] and$\displaystyle p_{2}$$\displaystyle are two distinct primes, then \displaystyle p_{1} \nmid p_{2} and \displaystyle \displaystyle p_{2}\nmid{p_{1}}, otherwise they couldn't possibly be primes. EDIT: I realised you said Mersenne numbers, not primes! God, what was I thinking? Perhaps it could be done by expanding \displaystyle \frac{2^a-1}{2^b-1} as \displaystyle \frac{(2-1)(2^{a-1}+2^{a-2}+\cdots + 2+1)}{(2-1)(2^{b-1}+2^{b-2}+\cdots + 2+1)} = \frac{(2^{a-1}+2^{a-2}+\cdots + 2+1)}{(2^{b-1}+2^{b-2}+\cdots + 2+1)}  or checking whether \displaystyle 2^b-1 can be written as \displaystyle 2ak+1$$$\displaystyle , for some [Math]k\in\mathbb{N}$. We need a trick, I think.

3. Hint:

1. Follows direclty from this (more generic) result
If $\displaystyle (m,n)$ = 1 then $\displaystyle (2^m - 1, 2^n-1)$ = 1

Can you prove the above result?

4. Well, I'm not sure how to prove that If (m,n) = 1 then (2^m - 1, 2^n - 1) = 1 .. Can you help me, please?