- Right. I'm mentioned this to him just about every time that

he posts a "proof" -- and he has never addressed the issue.

Every one of his proofs -- as far as I can tell -- claims that

such a construct can't exist. My counterexample shows that

such a construct can in fact exist, and the only reason it's

not an odd perfect number is because 22021 is divisible by 19.

But none of his arguments ever explain why such a construct

can exist with one of the factors composite, but cannot exist

with all of the factors prime.

On 7/11/2011 5:44 PM, djbroadhurst wrote:

>

>

> --- In primenumbers@yahoogroups.com,

> Jack Brennen<jfb@...> wrote:

>

>> According to your proof, why isn't N = 3^2*7^2*11^2*13^2*22021

>> an odd perfect number?

>

> Jack's point is well made; he has shown that

> sigma(N)/(2*N) = sigma(22021)/(1+22021).

> If 22021 were prime, N would be an odd perfect number.

> Of course, 22021 = 19^2*61 is not prime.

> However Bill did not mention decomposition into /primes/

> in his vague remarks; so he might not detect that N is abundant.

>

> David

>

>

>

> ------------------------------------

>

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> The Prime Pages : http://www.primepages.org/

>

> Yahoo! Groups Links

>

>

>

>

> - Thx, I will rework it.

________________________________

From: Tom Hadley <kctom99@...>

To: Mathieu Therrien <mathieu344@...>

Cc: "primenumbers@yahoogroups.com" <primenumbers@yahoogroups.com>

Sent: Thursday, July 14, 2011 4:41:26 PM

Subject: Re: [PrimeNumbers] Post-Cartesian Puzzle

Mathieu Therrien <mathieu344@...> wrote:

>

I think you have misunderstood the sigma() function. In Pari-GP, sigma(x) is the sum of the divisors of x. So sigma(9) = 1+3+9 = 13.

>If I understood the purpose of Sigma(N/M) = N/P_1 = M*P_2 for P = P_1 * P_2 correctly,

>

>then Many solution are possibles as long as (M+1) is divided by 2 only once

>

>for example u have m=5 ; N = P_2 * m * 3 = 15*P_2 and as long that P_2 is odd

>

>So m=5 and N=45 is 1 solution

>

>

The puzzle is: Find a pair of odd integers (N,m) with m|N,

sigma(N/m)*(1+m) = 2*N, and bigomega(m) = 1.

The proposed solution, N=45, m=5 doesn't work, since

sigma(45/5)*(1+5) = sigma(9)*6 = 13*6 = 78, which is not 2*N=2*45=90.

Tom Hadley

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