Re: why there are no Odd Perfect Numbers (not including 1) Re:

In article <1127194218.087978.38620@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
a_plutonium@xxxxxxxxxxx wrote:

> Thanks for the information, Gerry. Could you please answer some
> questions about the Chebyshev Theorem that says between N and 2N exists
> a prime. When did Chebyshev prove this theorem

1850, give or take 10 years. I'm sure this information is readily
available on the web.

> and briefly explain what
> mechanism went it proving that theorem.

Don't know, but there are some fairly simple proofs around.
There's one in Dan Cohen's undergrad combinatorics book.
There are probably more on the web, readily found by searching.

> And whether anyone has proved
> the Altered Chebyshev Theorem which says between N and 2N exists a
> special composite that has 2 and only 2 prime factors.

Well, suppose that between sqrt N and (sqrt 2) (sqrt N) there are
two primes, p and q. Then pq is the kind of number you want, and
it's between N and 2N.

Now it has been proved that if d is any positive real, and n is
large enough, then there's a prime between n and (1 + d) n.
Take d to be (4th root of 2) - 1 and you've got a prime between
n and (4th root of 2) n and another between (4th root of 2) n
and (sqrt 2) n, and you're done.

So the only thing missing here is the "large enough" part - how
large does n have to be to make the above go through? Again, that's
surely in the literature, and probably not hard to find on the web.
Once you've got it, it's just a finite search to show "Altered
Chebyshev" for all N.

> Note: Chris Heckman gave a nice short proof.

I must have missed it.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.