Quote:
Originally Posted by bhelmes
A peaceful and pleasant night for you,
if f>1 is the smallest factor of a Mersenne number, than there exists a n0>1 for the function f(n)=2n²1
where n0 is the minimum, so that f  f(n0) and exactly one prime g with f*g=f(n0) where g<f
Is this a correct and (well known) logical statement ?

Not true for Mersenne prime 2
^{3}  1 = 7: f = 7, n
_{0} = 3, n
_{0}^{2}  2 = f.
True for Mersenne primes greater than 7: If p > 3 is prime, and P = 2
^{p}  1 is prime, then f = P, and f divides (2^((p+1)/2))^2  2 = 2*f. Here, g = 2.
Note that for p > 3, 2*2^((p+1)/2) = 2^((p+3)/2) < 2^((p+p)/2) = 2^p, so 2^((p+1)/2) <= P/2 = f/2, whence n
_{0} = 2^((p+1)/2).
Not necessarily true for composite values of 2
^{p}  1, p prime. Example: p = 11, 2
^{11}  1 = 2047 = 23*89; f = 23, n
_{0} = 5, n
_{0}^{2}  2 = f. No prime g < f here.