Loading ...
Sorry, an error occurred while loading the content.

Re: [PrimeNumbers] The Function y^2 - x^2

Expand Messages
  • mikeoakes2@aol.com
    ... primes) which implies the functional equation thinking(Jens) thinking(Mike) -Mike Oakes [Non-text portions of this message have been removed]
    Message 1 of 6 , Feb 5, 2005
    • 0 Attachment
      >One snag:
      >y^2-x^2 = (y-x)*(y+x) is a product of 2 primes if and only if
      >(y-x and y+x are both primes) _or_ (y-x=1 and y+x is a product of 2
      primes)

      which implies the functional equation
      thinking(Jens) > thinking(Mike)

      -Mike Oakes


      [Non-text portions of this message have been removed]
    • Milton Brown
      Fermat realized this 300 years earlier. And, this is the basis of his factoring algorithm. Milton L. Brown miltbrown at earthlink.net ... conjecture is ...
      Message 2 of 6 , Feb 5, 2005
      • 0 Attachment
        Fermat realized this 300 years earlier.
        And, this is the basis of his factoring algorithm.

        Milton L. Brown
        miltbrown at earthlink.net

        > [Original Message]
        > From: Jens Kruse Andersen <jens.k.a@...>
        > To: <primenumbers@yahoogroups.com>
        > Date: 2/5/2005 6:52:19 AM
        > Subject: Re: [PrimeNumbers] The Function y^2 - x^2
        >
        >
        > Mike Oakes realized y^2-x^2 = (y-x)(y+x) and wrote:
        >
        > > If y^2-x^2 = p*q, where y > x >= 0, and p and q are prime, with p >= q,
        > > then
        > > y + x = p
        > > y - x = q
        > > so
        > > 2*y = p + q
        > > 2*x = p - q
        > >
        > > The first condition is always satisfiable ONLY if Goldbach's
        conjecture is
        > > true; and if it is satisfied then the second condition is trivially
        > > satisfied.
        >
        > One snag:
        > y^2-x^2 = (y-x)*(y+x) is a product of 2 primes if and only if
        > (y-x and y+x are both primes) _or_ (y-x=1 and y+x is a product of 2
        primes)
        >
        > The second part means x=y-1 is a solution if 2y-1 is a product of 2
        primes.
        > As example, let y=11. Then 2y-1 = 21 = 3*7, so x=10 is a solution.
        > There are of course also Goldbach solutions for y=11.
        >
        > This makes our problem slightly weaker than Goldbach's Conjecture, i.e.
        it is
        > conceivable (but highly unlikely) that Goldbach is false but there always
        is x
        > for our problem. That would be the case iff 2y-1 is a product of 2 primes
        for
        > all Goldbach counter examples 2y.
        >
        > As to why Goldbach remains unproved?
        > Well, lots and lots of conjectures with strong heuristic support are
        unproved.
        > Heuristics are usually useless for proofs.
        >
        > A probably weak explanation for why many prime conjectures are hard to
        prove:
        > Primes are defined by a property with multiplication.
        > Most conjectures involve addition (or subtraction).
        > Connecting these is easy in examples but often hard in proofs.
        >
        > --
        > Jens Kruse Andersen
        >
        >
        >
        >
        > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
        > The Prime Pages : http://www.primepages.org/
        >
        >
        > Yahoo! Groups Links
        >
        >
        >
        >
        >
        >
      Your message has been successfully submitted and would be delivered to recipients shortly.