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Re: [PrimeNumbers] Dead Pigeons (was perimetric prime sieve/generator = 2(p^(n-k)+p^k) )

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  • Alan Powell
    Bill Hmmm ... with reference to your conjecture below: Let us set e=2(p^(n-k)+p^k) then: For p=11, n=4, k=2 we have e=484 giving e-1=7*23 and e+1=5*97 For
    Message 1 of 9 , Jun 30, 2001
      Bill

      Hmmm ... with reference to your conjecture below:

      Let us set e=2(p^(n-k)+p^k) then:

      For p=11, n=4, k=2 we have e=484 giving e-1=7*23 and e+1=5*97

      For p=13, n=3, k=1 we have e=364 giving e-1=3*11^2 and e+1=5*73

      For p=13, n=3, k=2 we have e=364 giving e-1=3*11^2 and e+1=5*73

      For p=17, n=4, k=2 we have e=1156 giving e-1=3*5*7*11 and e+1=13*89

      For p=17, n=5, k=2 we have e=10404 giving e-1=101*103 and e+1=5*2081

      and so on .... in general an infinite pile of dead pigeons!


      These kinds of conjectures are best first validated with a few simple
      lines of Mathematica or Maple, for example:

      Do[p=Prime[pi];
      e=2(p^(n-k)+p^k);
      If[!(PrimeQ[e-1]||PrimeQ[e+1]),
      Print["p=",p," n=",n," k=",k," e=",e," e+1=",FactorInteger[e-1]," e-1=",FactorInteger[e+1]]];
      ,{pi,2,8},{n,1,pi-1},{k,1,n-1}];

      Regards

      Alan Powell
      Pigeon slayer

      At 03:07 AM 6/30/01, you wrote:
      >take a prime number (p), raise it to any power (n),
      >make a rectangle out of that number (with sides
      >p^(n-k) and p^k). Calculate the perimeter of that
      >rectangle. Add or subtract 1. At least one of these
      >will be a prime number ... I think. At any rate, all
      >the primes may be generated this way ... I think.
    • Bill Krys
      Dear Alan et al, can you find any for me that don t work for e=2(2^(n-k)+2^k)? Bill ... ===== Bill Krys Email: billkrys@yahoo.com Toronto, Canada (currently:
      Message 2 of 9 , Jun 30, 2001
        Dear Alan et al,

        can you find any for me that don't work for
        e=2(2^(n-k)+2^k)?

        Bill

        --- Alan Powell <powella@...> wrote:
        > Bill
        >
        > Hmmm ... with reference to your conjecture below:
        >
        > Let us set e=2(p^(n-k)+p^k) then:
        >
        > For p=11, n=4, k=2 we have e=484 giving e-1=7*23
        > and e+1=5*97
        >
        > For p=13, n=3, k=1 we have e=364 giving e-1=3*11^2
        > and e+1=5*73
        >
        > For p=13, n=3, k=2 we have e=364 giving e-1=3*11^2
        > and e+1=5*73
        >
        > For p=17, n=4, k=2 we have e=1156 giving
        > e-1=3*5*7*11 and e+1=13*89
        >
        > For p=17, n=5, k=2 we have e=10404 giving
        > e-1=101*103 and e+1=5*2081
        >
        > and so on .... in general an infinite pile of dead
        > pigeons!
        >
        >
        > These kinds of conjectures are best first validated
        > with a few simple
        > lines of Mathematica or Maple, for example:
        >
        > Do[p=Prime[pi];
        > e=2(p^(n-k)+p^k);
        > If[!(PrimeQ[e-1]||PrimeQ[e+1]),
        > Print["p=",p," n=",n," k=",k," e=",e,"
        > e+1=",FactorInteger[e-1],"
        > e-1=",FactorInteger[e+1]]];
        > ,{pi,2,8},{n,1,pi-1},{k,1,n-1}];
        >
        > Regards
        >
        > Alan Powell
        > Pigeon slayer
        >
        > At 03:07 AM 6/30/01, you wrote:
        > >take a prime number (p), raise it to any power (n),
        > >make a rectangle out of that number (with sides
        > >p^(n-k) and p^k). Calculate the perimeter of that
        > >rectangle. Add or subtract 1. At least one of these
        > >will be a prime number ... I think. At any rate,
        > all
        > >the primes may be generated this way ... I think.
        >
        >


        =====
        Bill Krys
        Email: billkrys@...
        Toronto, Canada (currently: Beijing, China)

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      • d.broadhurst@open.ac.uk
        ... at random: n=11, k=3: 2*(2^8+2^3)+1 trivially factors as: 23^2 2*(2^8+2^3)-1 trivially factors as: 17*31 with zillions more where those came from...
        Message 3 of 9 , Jun 30, 2001
          Bill Krys asked:
          > can you find any for me that don't work for
          > e=2(2^(n-k)+2^k)?
          at random: n=11, k=3:
          2*(2^8+2^3)+1 trivially factors as: 23^2
          2*(2^8+2^3)-1 trivially factors as: 17*31
          with zillions more where those came from...
          Sorry Bill, but the prime cicle is not so easily
          rectangulated:-)
          David
        • Bill Krys
          Okay, Dave, but what about the other half of the conjecture that all primes may be generated from 1 in such manner, despite creating many composites? Can you
          Message 4 of 9 , Jul 1, 2001
            Okay, Dave,

            but what about the other half of the conjecture that
            all primes may be generated from 1 in such manner,
            despite creating many composites? Can you find a
            counter example?

            Bill

            =====
            Bill Krys
            Email: billkrys@...
            Toronto, Canada (currently: Beijing, China)

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            Get personalized email addresses from Yahoo! Mail
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          • d.broadhurst@open.ac.uk
            ... Also false, I believe. Consider the prime q=103. I can see no way of writing either 51 or 52 in the form p^a+p^b. David (not Dave:-)
            Message 5 of 9 , Jul 1, 2001
              Bill Krys wrote:

              > but what about the other half of the conjecture that
              > all primes may be generated

              Also false, I believe. Consider the prime q=103.
              I can see no way of writing either 51 or 52
              in the form p^a+p^b.

              David (not Dave:-)
            • MICHAEL HARTLEY
              To: Alan Powell Copies to: primenumbers@yahoogroups.com From: Bill Krys Date sent:
              Message 6 of 9 , Jul 1, 2001
                To: Alan Powell <powella@...>
                Copies to: primenumbers@yahoogroups.com
                From: Bill Krys <billkrys@...>
                Date sent: Sat, 30 Jun 2001 06:16:28 -0700 (PDT)
                Subject: [PrimeNumbers] What about perimetric prime sieve/generator = 2(2^(n-k)+2^k) )

                > Dear Alan et al,
                >
                > can you find any for me that don't work for
                > e=2(2^(n-k)+2^k)?

                2*(2^(8-1)+2^1)-1 factor : 7
                2*(2^(8-1)+2^1)+1 factor : 3
                2*(2^(8-4)+2^4)-1 factor : 3
                2*(2^(8-4)+2^4)+1 factor : 5
                2*(2^(9-1)+2^1)-1 factor : 5
                2*(2^(9-1)+2^1)+1 factor : 11
                2*(2^(9-3)+2^3)-1 factor : 11
                2*(2^(9-3)+2^3)+1 factor : 5
                2*(2^(10-1)+2^1)-1 factor : 13
                2*(2^(10-1)+2^1)+1 factor : 3
                2*(2^(10-4)+2^4)-1 factor : 3
                2*(2^(10-4)+2^4)+1 factor : 7


                >
                > > At 03:07 AM 6/30/01, you wrote:
                > > >take a prime number (p), raise it to any power (n),
                > > >make a rectangle out of that number (with sides
                > > >p^(n-k) and p^k). Calculate the perimeter of that
                > > >rectangle. Add or subtract 1. At least one of these
                > > >will be a prime number ... I think. At any rate,
                > > all
                > > >the primes may be generated this way ... I think.
                > >
                > >
                >
                >
                > =====
                > Bill Krys
                > Email: billkrys@...
                > Toronto, Canada (currently: Beijing, China)
                >
                > __________________________________________________
                > Do You Yahoo!?
                > Get personalized email addresses from Yahoo! Mail
                > http://personal.mail.yahoo.com/
                >
                > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
                > The Prime Pages : http://www.primepages.org
                >
                >
                >
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                >
                >


                Michael Hartley : Michael.Hartley@...
                Head, Department of Information Technology,
                Sepang Institute of Technology
                +---Q-u-o-t-a-b-l-e---Q-u-o-t-e----------------------------------
                "If you entertain a thought, it becomes an attitude..."
              • MICHAEL HARTLEY
                To: d.broadhurst@open.ac.uk Copies to: primenumbers@yahoogroups.com From: Bill Krys Date sent: Sun, 1
                Message 7 of 9 , Jul 1, 2001
                  To: d.broadhurst@...
                  Copies to: primenumbers@yahoogroups.com
                  From: Bill Krys <billkrys@...>
                  Date sent: Sun, 1 Jul 2001 08:33:19 -0700 (PDT)
                  Subject: Re: [PrimeNumbers] Re: sieve = 2(p^(n-k)+p^k)

                  > Okay, Dave,
                  >
                  > but what about the other half of the conjecture that
                  > all primes may be generated from 1 in such manner,
                  > despite creating many composites? Can you find a
                  > counter example?

                  Yes. 103 cannot be written 2(p^k + p^(n-k))+/-1 for any prime p,
                  integers n >= k >= 0.

                  Proof: Left as an exercise to the reader.

                  Puzzle: Find the _next_ such prime...

                  [ PS - I almost embarrassed myself by saying "Yes. 31 cannot..." ]




                  >
                  > Bill
                  >
                  > =====
                  > Bill Krys
                  > Email: billkrys@...
                  > Toronto, Canada (currently: Beijing, China)
                  >
                  > __________________________________________________
                  > Do You Yahoo!?
                  > Get personalized email addresses from Yahoo! Mail
                  > http://personal.mail.yahoo.com/
                  >
                  > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
                  > The Prime Pages : http://www.primepages.org
                  >
                  >
                  >
                  > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
                  >
                  >


                  Michael Hartley : Michael.Hartley@...
                  Head, Department of Information Technology,
                  Sepang Institute of Technology
                  +---Q-u-o-t-a-b-l-e---Q-u-o-t-e----------------------------------
                  "If you entertain an attitude, it becomes an action..."
                • d.broadhurst@open.ac.uk
                  ... 103,139,151,157,199,223,233,239,241,307,311,313,353,367,373,379, 409,419,421,431,433,439,443,463,571,593,599,601,607,619,631,643,
                  Message 8 of 9 , Jul 1, 2001
                    > Puzzle: Find the _next_ such prime...
                    103,139,151,157,199,223,233,239,241,307,311,313,353,367,373,379,
                    409,419,421,431,433,439,443,463,571,593,599,601,607,619,631,643,
                    659,661,673,683,727,733,739,743,751,757,809,811,823,827,829,853,
                    857,859,877,883,911,919,941,947,953,967,991,997...
                  • d.broadhurst@open.ac.uk
                    Next puzzle: Why is EIS sequence ID Number: A033227 Sequence: 43,103*,139*,157*,181,277,367*,439*,523,547, 607*,673*,751*,823*,991*,997*... Name: Primes
                    Message 9 of 9 , Jul 1, 2001
                      Next puzzle: Why is EIS sequence

                      ID Number: A033227
                      Sequence: 43,103*,139*,157*,181,277,367*,439*,523,547,
                      607*,673*,751*,823*,991*,997*...
                      Name: Primes of form x^2+39*y^2.

                      such a fecund source of exceptions to Bill's conjecture?

                      [*] 11 of the first 16 elements are non-Krys
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