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

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  • Phil Carmody
    ... 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
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      --- "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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