In August 2008, the 45th known Mersenne prime, a mere 243,112,609-1 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 2N-1 is prime, then N must also be prime (Or contrapositively, if N is composite, then 2N-1 is also composite)
- if a number of the form 2N-1 is prime then the number 2(N-1) x (2N-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 2a 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