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2^a*3^b one away from a prime

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  • marku606
    Given any prime expressed as a+b, is there always some a,b such that 2^a*3^b is one away from a prime? I doubt it but have yet to find a counterexample. Below
    Message 1 of 3 , Nov 19, 2009
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      Given any prime expressed as a+b, is there always some a,b such that 2^a*3^b is one away from a prime?

      I doubt it but have yet to find a counterexample.

      Below are primes < 3187 which produced five or fewer solutions. [prime, number of times that 2^a*3^b is one away from a prime]

      [2, 2] [3, 3] [5, 5] [11, 5] [317, 5] [347, 3] [677, 4] [739, 4] [809, 5] [857, 5] [1033, 5] [1229, 5] [1291, 5] [1319, 5] [1451, 2] [1471, 5] [1663, 3] [1721, 5] [2069, 4] [2477, 5] [2659, 4] [2677, 3]

      Mark
    • Jens Kruse Andersen
      ... I expect infinitely many counter examples but there are none below 7500. I only computed one prime for each prime sum a+b. -- Jens Kruse Andersen
      Message 2 of 3 , Nov 19, 2009
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        Mark wrote:
        > Given any prime expressed as a+b, is there always some a,b
        > such that 2^a*3^b is one away from a prime?

        I expect infinitely many counter examples but there are none below 7500.
        I only computed one prime for each prime sum a+b.

        --
        Jens Kruse Andersen
      • marku606
        ... I think you re right Jens, infinitely many. By observation, the average number of solutions for a given prime seems to be roughly constant, around 10. For
        Message 3 of 3 , Nov 19, 2009
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          --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
          >
          > Mark wrote:
          > > Given any prime expressed as a+b, is there always some a,b
          > > such that 2^a*3^b is one away from a prime?
          >
          > I expect infinitely many counter examples but there are none below 7500.
          > I only computed one prime for each prime sum a+b.
          >

          I think you're right Jens, infinitely many.

          By observation, the average number of solutions for a given prime seems to be roughly constant, around 10.

          For instance the first 13 primes after 2000 yield the following number of hits:

          [2003, 8] [2011, 9] [2017, 7] [2027, 12] [2029, 10] [2039, 15] [2053, 7] [2063, 15] [2069, 4] [2081, 9] [2083, 6] [2087, 12] [2089, 10] average is 9.5

          The first 13 primes after 1000 yield the following:

          [1009, 13] [1013, 11] [1019, 10] [1021, 10] [1031, 10] [1033, 5] [1039, 7] [1049, 9] [1051, 13] [1061, 8] [1063, 9] [1069, 13] [1087, 10] average is 9.8

          The first 13 primes after 10 yield the following:

          [11,5] [13, 10] [17, 10] [19, 7] [23, 9] [29, 6] [31, 13] [37, 10] [41, 13] [43, 11] [47, 12] [53, 11] [59, 11] average is 9.8

          Alas I don't have any heuristic accurate enough to confirm or deny that the average will stay around 10. :)

          Nor am I savvy enough in statistics to deduce from the observed deviations a ballpark estimate as to when a zero would be expected.

          It's nice to see however so many primes nestled up against these "three smooth" numbers.

          Mark
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