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July 10th, 2019, 06:58 AM   #1
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From: Steyning, England

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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?
JonY is offline  
July 10th, 2019, 02:26 PM   #2
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Math Focus: primes of course
Try x=109
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July 10th, 2019, 11:11 PM   #3
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Yes JonY, your claim is known since Fermat (1640).

Let p be an odd prime, then any factor q of 2^p-1 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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