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Old 2018-01-10, 13:30   #3
Nick's Avatar
Dec 2012
The Netherlands

141110 Posts

As this is related to Mersenne numbers, let's turn it into an exercise for anyone interested.

1. Show for all positive integers m,n that if m divides n then \(2^m-1\) divides \(2^n-1\).

Let q be an odd prime number.
2. Show for all positive integers m,n with m>n that if q divides \(2^m-1\) and q divides \(2^n-1\) then q divides \(2^{m-n}-1\).

Let p be an odd prime number as well and suppose that q divides \(2^p-1\).
3. Show for all positive integers n that q divides \(2^n-1\) if and only if p divides n.
4. Conclude that \(q\equiv 1\pmod{p}\).
5. Show that 2 is a square in the integers modulo q, and conclude that \(q\equiv\pm1\pmod{8}\).
For the last part, the first example here may be useful.

Last fiddled with by Nick on 2018-01-10 at 16:09 Reason: Clarification
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