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Proth's theorem extended

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  • Bill Bouris
    Hello, let Q= k*2^n +1, k
    Message 1 of 7 , Jul 28 10:53 AM
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      Hello,
      let Q= k*2^n +1, k< 2^n +1 be odd, and 'n' be prime.
      Proth extended: if a^((Q-1)/4) == +/-1(mod Q), then 'Q' is prime.
      my rule: also, if a= 2 & n is prime, then Q will NEVER be pseudo-prime.

      extended portion:
      if m/is a natural number/, then every odd divisor 'q' of a^(2^(m+1)) +1 implies that q == 1(mod(a^(m+2))) [concluded from generalized Fermat-number 'proofs' by Proth himself but replacing 'm' with (m+1) in his original argument]. similarly, a^(2^(m+1)) -1 implies q == -1(mod(a^(m+2))) by the same observations.

      Now, if 'p' is any prime divisor of 'R', then a^((R-1)/4) = (a^k)*(a^(n-2)) == +/- 1(mod p) implies that either p == 1(mod(a^n)) or p == -1(mod (a^n)).

      Thus, if 'R' is composite, 'R' will be the product of at least two primes each of which has minimum value 2^n +1; quickly...

      2^n +1 > 0, 2^n > -1, k*2^n > -k, and k*2^n +1 > -k +1; so 2^n +1 would be the minimum choice instead of the factor... 2^n -1 > 0, 2^n > +1, k*2^n > +k, and k*2^n +1 > k +1 where both 'k's are odd; 2^n +1 would be minimal.

      it follow that...
      k*2^n +1 >= (2^n +1) * (2^n +1) = (2^n)*(2^n) + 2*(2^n) +1; 1's cancel so,
      k*(2^n) >= 2^n*(2^n) +2(2^n) and is k >= 2^n +2 is a contradiction. This was exactly how Proth found the boundary for 'k' except I had to rule out the other 'quickly...' supposed minimal factor choice.

      Hence, if a^((Q-1)/4) == +/-1(mod Q), then 'Q' is also prime for k< 2^n +1.
      *QED

      my rule: looking at Q-1 = 2^(2s)*f; if n = 2, k = 1 and Q= 1*2^2 + 1, and ((Q-1)/4) == 1(mod Q) is trival; but if n is and odd prime then Q-1 = (2^(2s))*f where f = 2*g and g= 2^n+1; so dividing by 4 means that f = 2^(2s)*(2^n) and upon examination of any psuedo-prime minus one... I can easily conclude that Q-1 = (only a product of 2's) isn't the correct composition for a pseudo-prime minus one.

      Hence, if a= 2 and n is prime, then Q will NEVER produce a pseudo-prime.

      does anyone concur... ??? wjb
    • leavemsg1
      I have extended Proth s theorem. It can be found on my website... www.oddperfectnumbers.com under the Other Short Proofs heading. You will also find the proof
      Message 2 of 7 , Apr 9, 2011
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        I have extended Proth's theorem.

        It can be found on my website...
        www.oddperfectnumbers.com under
        the Other Short Proofs heading.

        You will also find the proof for
        "why odd-perfect numbers don't
        exist" among others. Enjoy!

        Bill
      • Dimiter Skordev
        Hi, I am afraid that your idea why odd-perfect numbers don t exist would not work due to the existence of odd abundant numbers. Best regards, Dimiter
        Message 3 of 7 , Apr 11, 2011
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          Hi,

          I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.

          Best regards,
          Dimiter

          --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:
          >
          > I have extended Proth's theorem.
          >
          > It can be found on my website...
          > www.oddperfectnumbers.com under
          > the Other Short Proofs heading.
          >
          > You will also find the proof for
          > "why odd-perfect numbers don't
          > exist" among others. Enjoy!
          >
          > Bill
          >
        • leavemsg1
          Dimiter, first of all, you haven t even visited my website... and second, the argument is so simple that it does NOT require that someone even consider the
          Message 4 of 7 , Apr 11, 2011
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            Dimiter,

            first of all, you haven't even visited my website...
            and second, the argument is so simple that it does NOT
            require that someone even consider the summand portions
            which would make the answer deficient or abundant. The
            proof combines Euler's work w/that of Jacques Tochard's
            from 1953; I just threaded the ideas together. (prime,
            prime, prime; I have to mention them not to get spammed)
            Bill


            --- In primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@...> wrote:
            >
            > Hi,
            >
            > I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.
            >
            > Best regards,
            > Dimiter
            >
            > --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@> wrote:
            > >
            > > I have extended Proth's theorem.
            > >
            > > It can be found on my website...
            > > www.oddperfectnumbers.com under
            > > the Other Short Proofs heading.
            > >
            > > You will also find the proof for
            > > "why odd-perfect numbers don't
            > > exist" among others. Enjoy!
            > >
            > > Bill
            > >
            >
          • Jack Brennen
            Bill, please explain why solutions to: 2*P*Q^2 == (P+1)*sigma(Q^2), with P,Q odd relatively prime numbers 1 exist, but why none exist when P is prime.
            Message 5 of 7 , Apr 11, 2011
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              Bill, please explain why solutions to:

              2*P*Q^2 == (P+1)*sigma(Q^2),
              with P,Q odd relatively prime numbers > 1

              exist, but why none exist when P is prime.
              Because if one exists when P is prime, that
              would give an odd perfect number.

              (To see existence, try P=22021, Q=3003.)





              On 4/11/2011 1:43 PM, leavemsg1 wrote:
              > Dimiter,
              >
              > first of all, you haven't even visited my website...
              > and second, the argument is so simple that it does NOT
              > require that someone even consider the summand portions
              > which would make the answer deficient or abundant. The
              > proof combines Euler's work w/that of Jacques Tochard's
              > from 1953; I just threaded the ideas together. (prime,
              > prime, prime; I have to mention them not to get spammed)
              > Bill
              >
              >
              > --- In primenumbers@yahoogroups.com, "Dimiter Skordev"<skordev@...> wrote:
              >>
              >> Hi,
              >>
              >> I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.
              >>
              >> Best regards,
              >> Dimiter
              >>
              >> --- In primenumbers@yahoogroups.com, "leavemsg1"<leavemsg1@> wrote:
              >>>
              >>> I have extended Proth's theorem.
              >>>
              >>> It can be found on my website...
              >>> www.oddperfectnumbers.com under
              >>> the Other Short Proofs heading.
              >>>
              >>> You will also find the proof for
              >>> "why odd-perfect numbers don't
              >>> exist" among others. Enjoy!
              >>>
              >>> Bill
              >>>
              >>
              >
              >
              >
              >
              > ------------------------------------
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              > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
              > The Prime Pages : http://www.primepages.org/
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              >
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              >
              >
              >
            • Dimiter Skordev
              Of course I visited Bill s website, but I thought the author gives his explanations about the non-existence of odd perfect numbers in the target page of the
              Message 6 of 7 , Apr 11, 2011
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                Of course I visited Bill's website, but I thought the author gives his explanations about the non-existence of odd perfect numbers in the target page of the link "Odd Perfect Numbers Don't
                Exist" (it turned out this is just the root page of the site). The opinion I expressed in my previous message is based on the content of the above-mentioned page.

                --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:
                >
                > Dimiter,
                >
                > first of all, you haven't even visited my website...
                > and second, the argument is so simple that it does NOT
                > require that someone even consider the summand portions
                > which would make the answer deficient or abundant. The
                > proof combines Euler's work w/that of Jacques Tochard's
                > from 1953; I just threaded the ideas together. (prime,
                > prime, prime; I have to mention them not to get spammed)
                > Bill
                >
                >
                > --- In primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@> wrote:
                > >
                > > Hi,
                > >
                > > I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.
                > >
                > > Best regards,
                > > Dimiter
                > >
                > > --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@> wrote:
                > > >
                > > > I have extended Proth's theorem.
                > > >
                > > > It can be found on my website...
                > > > www.oddperfectnumbers.com under
                > > > the Other Short Proofs heading.
                > > >
                > > > You will also find the proof for
                > > > "why odd-perfect numbers don't
                > > > exist" among others. Enjoy!
                > > >
                > > > Bill
                > > >
                > >
                >
              • leavemsg1
                I m am sorry for accusing you of not visiting my website. I checked my Google Analytics program on the day of your visit , and it reported that no visits came
                Message 7 of 7 , Apr 13, 2011
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                  I'm am sorry for accusing you of not visiting my website.
                  I checked my Google Analytics program on the day of your
                  "visit", and it reported that no visits came in on that
                  day; I stand corrected! www.oddperfectnumbers.com

                  Bill

                  --- In primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@...> wrote:
                  >
                  > Of course I visited Bill's website, but I thought the author gives his explanations about the non-existence of odd perfect numbers in the target page of the link "Odd Perfect Numbers Don't
                  > Exist" (it turned out this is just the root page of the site). The opinion I expressed in my previous message is based on the content of the above-mentioned page.

                  > > It can be found on my website...
                  > > www.oddperfectnumbers.com under
                  > > the Other Short Proofs heading.

                  > > You will also find the proof for
                  > > "why odd-perfect numbers don't
                  > > exist" among others. Enjoy!
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