--On Wednesday, June 11, 2003 8:36 PM +0000 cite13083

<

cite13083@...> wrote:

> --- In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:

>> Is it not relatively simple to double check whether prime or not -

> simply

>> add the divisors and see if the result is 2n?

>

> Perhaps you've confused Mersenne primes with their associated perfect

> numbers.

Perfect numbers are never prime - all prime numbers are deficient. This is

a pretty stupid way to check for perfect prime numbers - especially since

it is likely that all perfect numbers are even, and with an exception even

numbers are not prime.

> Exercise: How many proper divisors does the 39th perfect number have?

I assume you are referring to the 39th known perfect number, which is the

39th, or possibly 40th or 41st, even perfect number. The number in

question is 2^13466916(2^13466917 - 1)

It has 13466917*2-1 proper divisors - we can have any number of factors of

2, from 0 to 13466916, either with or without the megaprime. Note that

when we have no 2's, and no megaprime, we have the perfectly valid divisor

of 1. When we have all possible factors, we have the number itself, thus

the -1.

As a simpler illustration, take the perfect number 2^2*(2^3-1). It has 2

proper divisors with and 3 without the prime

2^3-1 - respectively, 1, 2, 4 and 7, 14. Thus the total is 2*3-1 - again,

1 less than twice the original exponent.

Is this correct?

Nathan