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In August 2008, the 45th known Mersenne prime, a mere 2^43,112,609-1 was discovered by the Great Internet Mersenne Prime Search! Our puzzle this week is really just to rediscover for yourself proofs that 

• if a number of the form 2^N-1 is prime, then N must also be prime (Or contrapositively, if N is composite, then 2^N-1 is also composite)
• if a number of the form 2^N-1 is prime then the number 2^(N-1) x (2^N-1) is perfect— that is, it is the sum of all its proper divisors.

For example, 2³ – 1 = 7, which happens to be prime. 2² x (2³-1) = 28, which has proper divisors 1, 2, 4, 7, and 14, which sum to (drumroll) 28.

For fun you might look around for numbers of the form 2^{a prime} -1 that are not themselves prime; this shouldn’t take too long since these are far more common than that those that are, the Mersenne Primes.

If you want a little more of a challenge, try to prove that

• any even perfect number must be of this form

and if you want to be really famous, settle the conjectures that

• this takes care of everything—in other words that there are no odd proper numbers
• but that there are in fact infinitely many Mersenne primes and so infinitely many even perfect numbers