Quote:
Originally Posted by Nick
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 \(2^m1\) divides \(2^n1\).

I think this is sufficient?
NB Spoiler below.
If
Then, for example:
is an integer for all integer values of x and n.
And:
Generalised:
This is an integer for all integer values of n and x.
Now, if nm, then m = x * n, where m, n and x are all integers.
So:
Therefore:
As above:
It follows:
So:
And f(x) is an integer, so
divides
when n divides m.