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Prime Game

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  • Werner D. Sand
    7,31,109,349,1061,103,29,97,43,13 11,41,131,37 17,59 If you sum 3 consecutive odd prime numbers p,q,r, you get a number s which is either prime or not:
    Message 1 of 4 , Jan 2, 2006
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      7,31,109,349,1061,103,29,97,43,13
      11,41,131,37
      17,59

      If you sum 3 consecutive odd prime numbers p,q,r, you get a number s
      which is either prime or not: p+q+r=s. If s is prime, you call it p
      and repeat the game. If s is not prime, you call the greatest prime
      factor p and repeat the game. Finally, you get into an infinite
      cycle, which is one of the above 3 series, no matter what prime
      numbers you choose. Example: p=7

      7+11+13=31
      31+37+41=109
      109+113+127=349
      349+353+359=1061
      1061+1063+1069=3193=31*103
      103+107+109=319=11*29
      29+31+37=97
      97+101+103=301=7*43
      43+47+53=143=11*13
      13+17+19=49=7*7
      7+11+13…

      Questions:

      1.) Why doesn't the series diverge? Proof?
      2.) Why are there these 3 series and not 4 or 5? Proof?
      3.) Instead of s=p+q+r try s=p²+q²+r².
    • Kermit Rose
      From: Werner D. Sand Date: 01/02/06 09:30:35 To: primenumbers@yahoogroups.com Subject: [PrimeNumbers] Prime Game 7,31,109,349,1061,103,29,97,43,13 11,41,131,37
      Message 2 of 4 , Jan 2, 2006
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        From: Werner D. Sand
        Date: 01/02/06 09:30:35
        To: primenumbers@yahoogroups.com
        Subject: [PrimeNumbers] Prime Game

        7,31,109,349,1061,103,29,97,43,13
        11,41,131,37
        17,59

        If you sum 3 consecutive odd prime numbers p,q,r, you get a number s
        which is either prime or not: p+q+r=s. If s is prime, you call it p
        and repeat the game. If s is not prime, you call the greatest prime
        factor p and repeat the game. Finally, you get into an infinite
        cycle, which is one of the above 3 series, no matter what prime
        numbers you choose. Example: p=7

        7+11+13=31
        31+37+41=109
        109+113+127=349
        349+353+359=1061
        1061+1063+1069=3193=31*103
        103+107+109=319=11*29
        29+31+37=97
        97+101+103=301=7*43
        43+47+53=143=11*13
        13+17+19=49=7*7
        7+11+13…

        Questions:

        1.) Why doesn't the series diverge? Proof?


        Kermit says:

        I started with 113. How far would I have to go to end up with one of the
        3 series?

        I suspect that the termination into one of those three series has something
        to
        do with every prime >3 being of the form 6N+1 or 6N-1.

        [Non-text portions of this message have been removed]
      • Werner D. Sand
        Hi Kermit, after 3 steps you are in the 2nd cycle: 113+127+131=371=7*53 53+59+61=173 173+179+181=533=13*41 41... Werner
        Message 3 of 4 , Jan 3, 2006
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          Hi Kermit,

          after 3 steps you are in the 2nd cycle:

          113+127+131=371=7*53
          53+59+61=173
          173+179+181=533=13*41
          41...

          Werner
        • Werner D. Sand
          Guys, I call it a game , but it s mathematics!
          Message 4 of 4 , Jan 4, 2006
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            Guys, I call it a "game", but it's mathematics!
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