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Old 2021-09-03, 01:37   #2
Dr Sardonicus
 
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Feb 2017
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Quote:
Originally Posted by bhelmes View Post
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)=2n²-1
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 23 - 1 = 7: f = 7, n0 = 3, n02 - 2 = f.

True for Mersenne primes greater than 7: If p > 3 is prime, and P = 2p - 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 n0 = 2^((p+1)/2).

Not necessarily true for composite values of 2p - 1, p prime. Example: p = 11, 211 - 1 = 2047 = 23*89; f = 23, n0 = 5, n02 - 2 = f. No prime g < f here.
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