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

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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 1 of 9 , Jul 1 10:32 AM
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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 2 of 9 , Jul 1 7:27 PM
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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 3 of 9 , Jul 1 7:51 PM
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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 4 of 9 , Jul 1 8:19 PM
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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 5 of 9 , Jul 1 8:42 PM
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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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