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Certain Prime Integers with an Elegant Property

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  • w_sindelar@juno.com
    I discovered that certain primes have an elegant property. For convenience, call this type of prime Q. I think I can best explain what I mean by the following
    Message 1 of 2 , Dec 2, 2007
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      I discovered that certain primes have an elegant property. For
      convenience, call this type of prime Q. I think I can best explain what I
      mean by the following examples.

      Take the prime Q=439. The largest prime P smaller than Q is 433 and the
      smallest prime R larger than Q is 443.

      Write Q in positional notation as (400+30+9). Get the 3 primes
      immediately succeeding each of the terms (400, 30 and 9). They are (401,
      31 and 11). Their sum is 443, which equals the above R. Get the 3 primes
      immediately preceding each of the terms (400, 30 and 9). They are (397,
      29 and 7). Their sum is 433, which equals the above P. Notice the twin
      primes 29 and 31. As far as I went, at least one twin pair always showed
      up with primes of this type.

      The rule to follow in the above exercise is: If any term in the
      positional notation is 0, then the succeeding prime is 2 and the
      preceding prime is 0. If the term in the units position in the positional
      notation is 1, then the succeeding prime is 2 and the preceding prime is
      0.

      And there are Q primes where only the sum of the succeeding primes equals
      R. Here is an example of a chain of 4 Q's, namely (790733, 790739,
      790747, 790753), where the sum of the succeeding primes of the first term
      of the chain equals the second term, the sum of the succeeding primes of
      the second term equals the third term, etc. So far I found no chain of 5
      Q's.

      A particularly interesting example is this set of 4 consecutive Q primes
      in arithmetic progression: (23104127, 23104157, 23104187, 23104217). The
      sum of the succeeding primes of the first term of the CPAP equals the
      second term, the sum of the succeeding primes of the second term equals
      the third term, etc. I found no web reference to CPAP's with this
      property.

      And there are plenty of examples of Q primes where only the sum of the
      preceding primes equals P.

      Thanks folks for your time and any comments. Hope it proves
      recreationally interesting to someone.

      Bill Sindelar
    • w_sindelar@juno.com
      ... positional notation is 0, then the succeeding prime is 2 and the preceding prime is 0. If the term in the units position in the positional notation is 1,
      Message 2 of 2 , Dec 2, 2007
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        >Bill Sindelar wrote:
        >The rule to follow in the above exercise is: If any term in the
        positional notation is 0, then the succeeding prime is 2 and the
        preceding prime is 0. If the term in the units position in the positional
        notation is 1, then the succeeding prime is 2 and the preceding prime is
        0.

        Please add "If the term in the units position is 3, 5 or 7, the
        succeeding and preceding primes are 3, 5, 7 respectively".

        Bill
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