Ask A Scientist Mathematics Archive

Author:      kantars
How are perfect numbers generated?

Response #:  1 of 2
Author:      tee
If K is a prime and M(K) = 2^K-1 is also a prime (now called a Mersenne
prime) then P(K) =2^(K-1)*M(K) is a perfect number (the sum of all of its
proper divisors is equal to P(K)).  

Response #:  2 of 2
Author:      rcwinther
In fact, Euler proved that ALL *even* perfect numbers MUST be of the form
given in response #1.  Just recently a new Mersenne prime was found, thus
bringing the total number of known perfect numbers (if I remember correctly)
to 33. It is not known if there are infinitely many perfect numbers, nor it
is known whether there are any odd perfect numbers.  (However, in 1973 it
was proven that, if there are, they must be larger than 10^50.)