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July 10th, 2019, 06:58 AM   #1
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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?
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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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