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18883Re: R109297

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  • Mark Underwood
    Apr 23, 2007
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      --- In primenumbers@yahoogroups.com, "julienbenney" <jpbenney@...> wrote:
      > It's odd I just came to think of primes as I came in after the first
      rain in Melbourne for weeks.

      Glad to see your part of Australia is finally getting some rain. It's
      pretty dry down there I hear.

      I just noticed something in common with all the prime and probable
      prime repunits: R(2), R(19), R(23), R(317), R(1031), R(49081),
      R(86453) and R(109297).

      They are all prime of course, but all of their values mod (10^x) are
      also a prime, a power of 3 times a prime, or one.

      For instance take the largest prp prime repunit R(109297):

      109297 is prime
      109297 mod 100000 = 9297 = 3^2 * 1033
      109297 mod 10000 = 9297 = 3^2 * 1033
      109297 mod 1000 = 297 = 3^3 * 11
      109297 mod 100 = 97
      109297 mod 10 = 7

      Furthermore, notice that with the derived 1033 prime factor above, it
      too has all its values mod 10^x as prime, a power of 3 times a prime,
      or one.

      And so on with the rest of the prime repunits. Coincidence? I doubt
      it would hold for all larger prime repunits, but it's fun observing
      what is likely the law of small numbers in effect.

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