Question

Numbers of the form M(n) = 2^(n) - 1 are called Mersenne numbers. Prove that if p and q=2p+1 are both primes, then either q | M(p) or
q | (M(p) + 2 ), but not both.

Answer 1

Any suggestions as to how to approach this proof would be greatly appreciated as well.

Answer 2

I'd start by looking at sums of geometric series. In particular you know that \[(x-1)|(x^{n}-1)\]and\[(x+1)|(-x)^{n} -1\]If you haven't learned these in class, you might need to prove this.

Answer 3

Thank you! I haven't actually learned these in class, but I'll google them. :)

Answer 4

A word of warning, I'm not exactly sure how to prove this either, and I don't seem to be getting anywhere with that method. Rather, after looking around a little bit, I'd suggest distinguishing between primes p where \[p\equiv 3\;\;\;(\!\!\!\!\mod 4)\]vs where \[p\equiv 1\;\;\;(\!\!\!\mod 4)\]

Answer 5

Have you learned of the legendre symbol yet?

Answer 6

Sry! I stepped away for a bit. No, I haven't learned that yet.

Answer 7

I've learned a bit about congruences, though.

Answer 8

I was asking because I found a really short proof in a textbook I have, but it uses the legendre symbol. But everything I've seen suggests doing it mod 3 and mod 1.

Answer 9

Ok. I'll see what I can do. Btw, would you be ok with letting me know the name of the textbook, if it's alright with you?

Answer 10

If not, it's ok. :)

Answer 11

Elementary Theory of Numbers by Sierpinski. You can find it online for free.

Answer 12

The proof is at the beginning of chapter 10.

Answer 13

Thank you very much! I appreciate it. :)

Answer 14

you're welcome.