Quote:
Originally Posted by bhelmes
If f  Mp and p is prime then
a) f  2n²1
b) but not f=2n²1
a) is clear for me but b) is a guess (I checked some numbers)
The first and fastest counterexample is welcome (Or a proof ?)

First example: Taking n = 1, f = 1 divides 2
^{p}  1 for any p.
First example with prime f: p = 3, n = 2; f = 2*2
^{2}  1 = 7 divides 2
^{3}  1 = 7.
Second example with prime f: p = 5, n = 4; f = 2*4
^{2}  1 = 31 divides 2
^{5}  1 = 31
48 other examples known where f = 2
^{p}  1 is prime.
Hey, you didn't say f had to be a
proper divisor  or prime.
I checked the factor table of 2
^{n}  1, odd n < 1200, and did not find any examples with f prime other than f = M
_{p}
I didn't check for composite proper factors of the form 2*n
^{2}  1.