[primeform] quasi-perfect numbers fall with 2-line proof

leavemsg1@xxxxxxxxx

don't get mad... like someone else did.

why search endlessly for an example... when you can disprove it!

just use a pencil and paper to follow the algegra; it's simple.

I read in some forgotten article that if there exists a quasi-

perfect number, then 'n' would have to be an odd perfect square.

I'm using the convention that when sigma(n) = n that 'n' is a

perfect number and not the usual sigma(n) = 2n +1; don't be up-

set, if I redefine sigma's definition, since it could be equally

shown if I had chosen the more popular definition verses mine.

notably, if sigma(n^2)= n^2 +1 for an odd 'n'>3, then 'n^2' is

considered to be a quasi-perfect number! (plese excuse the early

substitution of n^2 in for 'n' already... it reads better, later).

so... for some odd 'n'> 3, (double-spaced to make it read better),

if sigma(n^2)= n^2 +1, then sigma(n^2) > (n+1)(n-1)= n^2 -1; taking

sigma AGAIN leads to... sigma(sigma(n^2)) > sigma(n^2 -1); which re-

duces to sigma(n^2 +1) > sigma(n^2 -1) using the sigma(n^2)= n^2 +1

definition again; I'm only interested in the 'proper' divisors and

am not including the 'improper' divisors into the sum.

without any hesitation, (n^2 -1) has the form '8m' and (n^2 +1) has

the form '8m +2' for the same 'm'; so, clearly, sigma(n^2 -1) is

more abundant than sigma(n^2 +1), but only when an odd 'n'> 3.

look...

if 'n' equals 5 and n^2 = 25, then... naturally...

sigma(26) is definitely < sigma(24) where 26 = 8m +2 and 24 = 8m

like it or not, this will be the case every time!

again...

if 'n' equals 7 and n^2 = 49, then...

sigma(50) is definitely < sigma(48) where 50 = 8m +2 and 48 = 8m

when I think about it, I don't even have to do the summation; whew!

hence, by contradiction, a quasi-perfect number cannot exist.

a mysterious 2-line proof; I would have made it longer, but taking

sigma on both sides AGAIN made the mystery... ... disappear!

the sigma I chose only deals with the 'proper' divisors; that's why

other professionals didn't catch the quasi's
.

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