--- "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

=====

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