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

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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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