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Re: [PrimeNumbers] Re: sieve = 2(p^(n-k)+p^k)

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  • 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 1 of 9 , Jul 1, 2001
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      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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    • 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 2 of 9 , Jul 1, 2001
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        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 3 of 9 , Jul 1, 2001
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          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)
          >
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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 4 of 9 , Jul 1, 2001
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            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 5 of 9 , Jul 1, 2001
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              > 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 6 of 9 , Jul 1, 2001
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                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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