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20125Re: [PrimeNumbers] Re: 2^m+3^n and 2^n+3^m

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  • Phil Carmody
    Apr 18, 2009
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      --- On Fri, 4/17/09, David Broadhurst <d.broadhurst@...> wrote:
      > Submitted to OEIS:
      >
      > Numbers n such that 2^x + 3^y is never prime when max(x,y) = n
      >
      > 1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 6578,
      > 7251, 7406, 7642, 8218, 8331, 9475, 9578, 9749
      >
      > Mark Underwood found that for each non-negative integer n < 1421
      > there is at least one prime of the form 2^m + 3^n or 2^n + 3^m
      > with m not exceeding n.
      >
      > This sequence consists of numbers for which there is no
      > such prime.
      >
      > David Broadhurst estimated that a fraction in excess of 1/800
      > of the natural numbers belongs to this sequence and found
      > 17 instances with n < 10^4.
      >
      > For each of the remaining 9983 non-negative integers n < 10^4,
      > a prime or probable prime of the form 2^x + 3^y was found with
      > max(x,y) = n.
      >
      > Each probable prime was subjected to a combination of
      > strong Fermat and strong Lucas tests.
      >
      > http://physics.open.ac.uk/~dbroadhu/cert/marktest.txt

      This form invites possibly the most bizarre, and remarkably efficient, sieve algorithm I've yet had the misfortune of considering. Good job I'm not coding currently... I'm going to be a tortured soul for at least 2 days until I forget about it.

      Phil
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