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Reflection in the next prime

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  • Zak Seidov
    Reflection in the next prime Take any n and next prime p , that is the smallest prime p n, reflect n in p and get a(n)=2p-n; then starting with n=1, we have
    Message 1 of 4 , Jul 31, 2003
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      Reflection in the next prime

      Take any n and "next prime p", that is the smallest prime
      p>n, "reflect" n in p and get a(n)=2p-n;
      then starting with n=1, we have the sequence a(n):
      3,4,7,6,9,8,15,14,13,12,15,14,21,20,19,18,21,20,27,26,25,24,35
      (A087030?)
      Now introduce b(n)=1/0 if a(n)is prime/composite:
      1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,
      0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,
      0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0(A087032?)
      Cases of prime a(n) or b(n)=1 are sure less numerous than b(n)=0. Two
      (trivial?) observations:
      there is no two subsequent ones; number c(n) of zeros in each group
      is odd:
      1,5,5,11,3,1,5,7,9,3,1,5,5,15,1,5,5,9,1,5,11,5,5,3,7,7,9,5,1,3,5,3,13,
      3,1,5,11,7,5,5,9,3,1,5,17,3,1,5,15,15,3,15,5,7,17,23,5(A087033?)
      Up to n=100,000, the maximal number of zeros in one group is 75
      (starting from which n? - i don't know!).
      But what about larger c(n):
      what is the longest series of subsequent n such that "reflection of n
      in the next prime p", 2p-n, is composite, p being the smallest prime
      >n?
      zak
    • Jose Ramón Brox
      Hey Zack: There can not be two ones consecutives because if n is even, 2p-n is even... the same fact explains the oddity of the zeroes chains (groups as you
      Message 2 of 4 , Jul 31, 2003
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        Hey Zack:

        There can not be two ones consecutives because if n is even, 2p-n is even... the same fact explains the oddity of the zeroes chains (groups as you call them): if n is even then n+2k is even too, n+(2k-1) could be prime, but 2*nextprime(n+2k) - (n+2k) is certainly composite, so the number c(n) must be odd.

        Jose Brox

        ----- Original Message -----
        From: Zak Seidov
        To: primenumbers@yahoogroups.com
        Sent: Thursday, July 31, 2003 8:29 PM
        Subject: [PrimeNumbers] Reflection in the next prime


        Reflection in the next prime

        Take any n and "next prime p", that is the smallest prime
        p>n, "reflect" n in p and get a(n)=2p-n;
        then starting with n=1, we have the sequence a(n):
        3,4,7,6,9,8,15,14,13,12,15,14,21,20,19,18,21,20,27,26,25,24,35
        (A087030?)
        Now introduce b(n)=1/0 if a(n)is prime/composite:
        1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,
        0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,
        0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0(A087032?)
        Cases of prime a(n) or b(n)=1 are sure less numerous than b(n)=0. Two
        (trivial?) observations:
        there is no two subsequent ones; number c(n) of zeros in each group
        is odd:
        1,5,5,11,3,1,5,7,9,3,1,5,5,15,1,5,5,9,1,5,11,5,5,3,7,7,9,5,1,3,5,3,13,
        3,1,5,11,7,5,5,9,3,1,5,17,3,1,5,15,15,3,15,5,7,17,23,5(A087033?)
        Up to n=100,000, the maximal number of zeros in one group is 75
        (starting from which n? - i don't know!).
        But what about larger c(n):
        what is the longest series of subsequent n such that "reflection of n
        in the next prime p", 2p-n, is composite, p being the smallest prime
        >n?
        zak


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      • Jon Perry
        even (xcus the pun) more fun, take Ulam s Spiral, and place 4 infinite mirrors on it, horizontal, vertical and both diagonals, which primes have the highest
        Message 3 of 4 , Jul 31, 2003
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          'even' (xcus the pun) more fun, take Ulam's Spiral, and place 4 infinite
          mirrors on it, horizontal, vertical and both diagonals, which primes have
          the highest reflective count?

          Jon Perry
          perry@...
          http://www.users.globalnet.co.uk/~perry/maths/
          http://www.users.globalnet.co.uk/~perry/DIVMenu/
          BrainBench MVP for HTML and JavaScript
          http://www.brainbench.com
        • Jose Ramón Brox
          Should have written 2*nextprime(n+(2k-1)) - n+(2k-1) could be prime instead of n+(2k-1)... Jose ... From: Jose Ramón Brox To: Prime Numbers Sent:
          Message 4 of 4 , Jul 31, 2003
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            Should have written " 2*nextprime(n+(2k-1)) - n+(2k-1) could be prime " instead of " n+(2k-1)..."

            Jose

            ----- Original Message -----
            From: Jose Ramón Brox
            To: Prime Numbers
            Sent: Thursday, July 31, 2003 9:34 PM
            Subject: Re: [PrimeNumbers] Reflection in the next prime


            Hey Zack:

            There can not be two ones consecutives because if n is even, 2p-n is even... the same fact explains the oddity of the zeroes chains (groups as you call them): if n is even then n+2k is even too, n+(2k-1) could be prime, but 2*nextprime(n+2k) - (n+2k) is certainly composite, so the number c(n) must be odd.

            Jose Brox

            ----- Original Message -----
            From: Zak Seidov
            To: primenumbers@yahoogroups.com
            Sent: Thursday, July 31, 2003 8:29 PM
            Subject: [PrimeNumbers] Reflection in the next prime


            Reflection in the next prime

            Take any n and "next prime p", that is the smallest prime
            p>n, "reflect" n in p and get a(n)=2p-n;
            then starting with n=1, we have the sequence a(n):
            3,4,7,6,9,8,15,14,13,12,15,14,21,20,19,18,21,20,27,26,25,24,35
            (A087030?)
            Now introduce b(n)=1/0 if a(n)is prime/composite:
            1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,
            0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,
            0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0(A087032?)
            Cases of prime a(n) or b(n)=1 are sure less numerous than b(n)=0. Two
            (trivial?) observations:
            there is no two subsequent ones; number c(n) of zeros in each group
            is odd:
            1,5,5,11,3,1,5,7,9,3,1,5,5,15,1,5,5,9,1,5,11,5,5,3,7,7,9,5,1,3,5,3,13,
            3,1,5,11,7,5,5,9,3,1,5,17,3,1,5,15,15,3,15,5,7,17,23,5(A087033?)
            Up to n=100,000, the maximal number of zeros in one group is 75
            (starting from which n? - i don't know!).
            But what about larger c(n):
            what is the longest series of subsequent n such that "reflection of n
            in the next prime p", 2p-n, is composite, p being the smallest prime
            >n?
            zak


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            [Non-text portions of this message have been removed]


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