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24369Re: Impossible to Prove ??

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  • Aldrich
    Aug 6 5:28 PM
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      > > I think its status will remain
      > > 'empirical observation' for quite some time. Since
      > > the qfbsolve procedure presupposes that it is true,
      > > and the good doctor declined to comment upon it.....
      >
      > There is no "presupposing" about qfbsolve(). The good doctor
      > David did provide a hint that qfbsolve() works for ALL primes
      > P of the form described above and also provided a pointer to
      > the relevant book. Considering that Pari/GP is open-source,
      > you could also look at the guts of qfbsolve(), see how (easy)
      > and why (difficult) it works with your own eyes :-)
      >

      In Philosophy, the term "presuppose" generally means that the
      truth of one Proposition depends partly on another that is not
      explicitly mentioned, so qfbsolve definitely does necessarily presuppose something about infinities of primes in the structure
      of binary quadratic forms in its methods. Lack of deductive
      proof for these underlying assumptions introduces a measure of
      doubt.

      > >
      > > As to the status of proofs of conjectures relating
      > > to qfbsolve, my guess is that many of them are still
      > > empirical.

      >
      > Nope. The qfbsolve algorithm works and there is a proof
      > that it works for all primes and all quadratic forms (giving
      > you either the representation or a telling you with
      > certainty that one doesn't exist). >
      > So no, there is no "still empirical" here -- the existence
      > has been proved constructively (via a resonably fast algorithm).
      >

      Although I believe in the reliability of Qfbsolve almost as
      much I believe in Goldbach's Conjecture, both are actually
      empirical observations until deductive proofs can be found.

      I think you might be surprised at David's opinion on the
      topic unless he has recently recanted. A few years ago he
      made a point about the desirability of deductive over inductive
      proof when he constructed a sequence that was all +1 mod 10
      factors until about the 10^80 mark, where he had placed a square
      of a -1 mod 10 factor. The lack of a counter-example simply
      is not proof of truth (old school).

      Aldrich
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