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

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  • MICHAEL HARTLEY
    To: d.broadhurst@open.ac.uk Copies to: primenumbers@yahoogroups.com From: Bill Krys Date sent: Sun, 1
    Message 1 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)
      >
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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 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 2 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 3 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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