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What is the formula for the 45th perfect number??P.S You need to know the 45th Merssenne prime in Euliced's

Formula

Guest Jun 1, 2014

Best Answer 

 #1
avatar+88980 
+5

Euclid proved that when 2p -1 is prime, then 2p-1(2p -1) is a perfect number. Such a prime is known as a Mersenne prime.

For 2p-1 to be prime, it is necessary that p be prime. (It doesn't work the other way....p might be prime, but 2p-1 might not be.)

The 45th perfect number - discovered in 2008 - has over 22 million digits - obviously impractical to write that out, here !! (Note that this is the 45th "ranked" perfect number......this "rank" may change if smaller perfect numbers are discovered. It is known that this would have to occur after the 43rd perfect number.......thus, the "ranking" of the first 43 is correct.)

The Mersenne prime that generated the 45th perfect number is (237,156,667 - 1). So, the "formula" would be:

237,156,667 -1 (237,156,667 - 1)

 

BTW....all the perfect numbers discovered so far are even. It is not known if any odd ones exist. If they do, they are greater than 101500 . (No...I don't know how this was proved !!)

   
CPhill  Jun 1, 2014
 #1
avatar+88980 
+5
Best Answer

Euclid proved that when 2p -1 is prime, then 2p-1(2p -1) is a perfect number. Such a prime is known as a Mersenne prime.

For 2p-1 to be prime, it is necessary that p be prime. (It doesn't work the other way....p might be prime, but 2p-1 might not be.)

The 45th perfect number - discovered in 2008 - has over 22 million digits - obviously impractical to write that out, here !! (Note that this is the 45th "ranked" perfect number......this "rank" may change if smaller perfect numbers are discovered. It is known that this would have to occur after the 43rd perfect number.......thus, the "ranking" of the first 43 is correct.)

The Mersenne prime that generated the 45th perfect number is (237,156,667 - 1). So, the "formula" would be:

237,156,667 -1 (237,156,667 - 1)

 

BTW....all the perfect numbers discovered so far are even. It is not known if any odd ones exist. If they do, they are greater than 101500 . (No...I don't know how this was proved !!)

   
CPhill  Jun 1, 2014

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