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24506Re: [PrimeNumbers] n, 2n-1, 2n+1 all prime or prime-power (maybe n-2 also)

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  • Jack Brennen
    Oct 1, 2012
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      First, you missed an easy one:

      4, 7, 9

      Second, the next one seems to be:

      (3^541-1)/2, 3^541-2, 3^541

      As far as the conjecture about the small examples with double
      powers being the only ones, that would seem to be related to
      the ABC Conjecture.



      On 10/1/2012 9:33 AM, WarrenS wrote:
      > If we demand that n, 2n-1, and 2n+1 all simultaneously be prime or prime-power,
      > then initial examples found by hand are
      >
      > n, 2n-1, 2n+1
      > 2, 3, 5
      > 3, 5, 7
      > 5, 9*, 11
      > 9*, 17, 19
      > 13, 25*, 27*
      > 41, 81*, 83
      > 121*,241,243*
      > (next one over 10^12)
      >
      > where * for prime powers.
      > You can easily see that is is impossible for all three to be prime (except in the
      > first two lines) since at least one member of the troika must be divisible by 3. Hence the only way to accomplish it, is to make that one be a power of 3.
      > So then we necessarily have an exponentially-sparse set of primes, in that sense resembling the famous "Mersenne primes" 2^p-1 as well as "Proth primes" like 3*2^n+1.
      > Another kind of resemblance is the fact that if Q=3^k then
      > Q+2 will be easy to test for primality if we know (Q+1)/2 is prime [since then it
      > would be a "safeprime"]. In the other direction, (Q-1)/2 might also be easy to test for primality...
      >
      > Perhaps the 13,25,27 and 121,241,243 lines are the only ones featuring a
      > a double (or triple) *.
      >
      > It seems to me that this is an interesting class of primes.
      > Perhaps even more interesting are the "pack of four" cases where n-2 is ALSO prime or prime-power.
      >
      >
      >
      >
      >
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