## Re: [PrimeNumbers] Two primes between P(n) and 2* P(n)

Expand Messages
• ... Are you assuming the truth of the hypothesis you re trying to prove? ... Some statements can be used to verify their own falsity P |- ~P and these
Message 1 of 2 , Jun 26, 2002
--- "John W. Nicholson" <johnw.nicholson@...> wrote:
> From Paulo Ribenboim's The Little Book of Big Primes (c) 1991
>
> The following is results by Ishikawa (1934) are also consequences
> of Tschebycheff's Theorems (see Trost's book):
> If n >= 2, then P(n) + P(n+1) > P(n+2);
> if m,n >=1, then P(m)*P(n) > P(m+n).
>
> Proof:
>
> If g(n) = P(n+1) - P(n) then we can rewrite this statement as
> 2*P(n) + g(n) > P(n) + g(n) + g(n+1)

Are you assuming the truth of the hypothesis you're trying to prove?

> or P(n) > g(n+1)
> and because P(n) + g(n+1) < 2*P(n) there are two primes < 2*P(n).
>
> QED

Some statements can be used to verify their own falsity
P |- ~P
and these statements can be used as contrapositives, thus to prove
~P. (All false statements can do this, as False |- Anything)

However, you can't use an assumption to prove itself as
for all P, P |- P

Phil

=====
--
"One cannot delete the Web browser from KDE without
losing the ability to manage files on the user's own
hard disk." - Prof. Stuart E Madnick, MIT.
So called "expert" witness for Microsoft. 2002/05/02

__________________________________________________
Do You Yahoo!?
Yahoo! - Official partner of 2002 FIFA World Cup
http://fifaworldcup.yahoo.com
Your message has been successfully submitted and would be delivered to recipients shortly.