An odd perfect number, N, would have to be of the form

N = p1^a1 p2^a2 . . . p_k^a_k

where each of the p's are odd primes.

The sum of the divisors of N, ( including N) is the product

S = (1 + p1 + p1^2 + . . . + p1^a1) ( 1 + p2 + p2^2 + . . . + p2^a2) .

. . (1 + p_k + p_k ^2 + . . . + p_k ^a_k )

Thus an odd perfect number must satisfy the requirement

2 p1^a1 p2^a2 . . . p_k^a_k = (1 + p1 + p1^2 + . . . + p1^a1) ( 1 +

p2 + p2^2 + . . . + p2^a2) . . . (1 + p_k + p_k ^2 + . . . + p_k ^a_k )

where each p1, p2, . . . p_k is odd and greater than 2.

Thus (1 + p1 + p1^2 + . . . + p1^a1) ( 1 + p2 + p2^2 + . . . + p2^a2)

. . . (1 + p_k + p_k ^2 + . . . + p_k ^a_k ) / [ p1^a1 p2^a2 . . .

p_k^a_k ] = 2

Is it possible for (1 + p1 + p1^2 + . . . + p1^a1) ( 1 + p2 + p2^2 + .

. . + p2^a2) . . . (1 + p_k + p_k ^2 + . . . + p_k ^a_k ) / [

p1^a1 p2^a2 . . . p_k^a_k ]

to equal 2 when all the p1, p2, ... p_k are odd and greater than 2?

Since 2 = 2 mod 4, we see that the numerator must not = 0 mod 4, and so

exactly one of the a1, a2, ...a_k must be odd, and all the rest must be

even.

This shows that an odd perfect number must be an odd power of a prime

times a square.

I'm sure that many more elementary observations have been made about

requirements of odd perfect numbers.

I anticipate that the proof that odd perfect numbers do not exist would

follow closely the proof that even perfect numbers are a prime times a

power of 2.

I have not yet done a literature research on it. Does anyone here know

off the top of their head other tidbits about odd perfect numbers?

Kermit <

kermit@... >