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20062Re: 2^m+3^n and 2^n+3^m

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  • David Broadhurst
    Apr 11 6:48 AM
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      --- In primenumbers@yahoogroups.com,
      "David Broadhurst" <d.broadhurst@...> wrote:

      > "Numbers n such that there are no primes of the forms
      > 2^m+3^n or 2^n+3^m for m < n"

      These numbers include

      1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 7251, 8218

      but my coverage of the range n < 8218 is not complete,
      so this is not yet a bona fide sequence.

      So far, I have tested more than 7000 values of n < 8300
      and found these 10 examples.

      This agrees tolerably well with my prior heuristics.

      To make a rough estimate of the density of primes of the
      form N = 2^a + 3^b, I sieved for prime factors < 10^5,
      with a and b running from 5000 to 5200, and found that
      5558 values of N survived. So I estimated the
      probability of primality to be C/log(N), with

      C =~ exp(Euler)*log(10^5)*5558/201^2 =~ 2.82.

      Hence I guessed that the probability of finding that
      a number n lies in this sequence exceeds

      exp(-C/log(2))*exp(-C/log(3)) > 1/800.

      I say "exceeds", since 2^n+3^m starts at size
      O(2^n) and ends at size O(3^n), as m runs from
      1 to n-1, while 2^m+3^n stays at size O(3^n).

      David
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