July 10th, 2019, 06:58 AM  #1 
Newbie Joined: May 2019 From: Steyning, England Posts: 1 Thanks: 0  Mersenne Primes
Mersenne Primes are prime numbers derived from the formulae (2**x)1, but this only works when x is also a prime number. However, this is not always the case, for example, when x = 11 the resultant number (2,047) is not prime, because 2047 has prime factors of 23 and 89, which are multiples of the power, (i.e. 11) plus 1 (11* 2 +1 = 23 and 11* 8 + 1 = 89). My claim, is that for all prime powers which do not generate a prime number will always have its prime factors as multiples of the power plus 1. Is this known? Can anyone direct me to some reading material if it is?

July 10th, 2019, 02:26 PM  #2 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 354 Thanks: 7 Math Focus: primes of course 
Try x=109

July 10th, 2019, 11:11 PM  #3 
Member Joined: Oct 2013 Posts: 60 Thanks: 6 
Yes JonY, your claim is known since Fermat (1640). Let p be an odd prime, then any factor q of 2^p1 must be of the form 2kp+1. Furthermore, q must be 1 or 7 mod 8, a theorem Euler (1750) discovered. Millions of factors of Mersenne numbers have been found with these two theorems. For more Information, see The Math behind GIMPS. Here is a proof of the theorems. $$$$ Last edited by skipjack; July 11th, 2019 at 12:51 AM. 

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