Re: [PrimeNumbers] Two primes between P(n) and 2* P(n)
- --- "John W. Nicholson" <johnw.nicholson@...> wrote:
> From Paulo Ribenboim's The Little Book of Big Primes (c) 1991Are you assuming the truth of the hypothesis you're trying to prove?
> 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).
> 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)
> or P(n) > g(n+1)Some statements can be used to verify their own falsity
> and because P(n) + g(n+1) < 2*P(n) there are two primes < 2*P(n).
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
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