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

#1**+5 **

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

For 2^{p}-1 to be prime, it is necessary that p be prime. (It doesn't work the other way....p might be prime, but 2^{p}-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 (2^{37,156,667} - 1). So, the "formula" would be:

2^{37,156,667 -1} (2^{37,156,667} - 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 10^{1500} . (No...I don't know how this was proved !!)

CPhill
Jun 1, 2014

#1**+5 **

Best Answer

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

For 2^{p}-1 to be prime, it is necessary that p be prime. (It doesn't work the other way....p might be prime, but 2^{p}-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 (2^{37,156,667} - 1). So, the "formula" would be:

2^{37,156,667 -1} (2^{37,156,667} - 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 10^{1500} . (No...I don't know how this was proved !!)

CPhill
Jun 1, 2014