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 itsproper divisors is equal to P(K)).
ALSO, Euler proved that ALL *even* perfect numbers MUST be of the form given above. 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.)
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) then 
(the sum of all of itsproper divisors is equal to P(K)).
. 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, 
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