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Re: prime +/- 2^n = primes: A tight sequence and a potential conjecture?

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  • Mark Underwood
    Thank you Jens! So 1973 is the first prime to generate no other primes in this fashion. And to think I always liked the 1970s. :) There are 42183 0 s below 50
    Message 1 of 3 , Jun 28, 2003
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      Thank you Jens! So 1973 is the first prime to generate no other
      primes in this fashion. And to think I always liked the 1970s. :)

      There are 42183 0's below 50 million? Egad. Somehow it has lost the
      specialness it had up to 313. :)

      Very interesting indeed, thanks again
      Mark



      --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
      <jens.k.a@g...> wrote:
      > Mark Underwood wrote:
      > > I was looking at the first twenty or so primes in binary notation
      for
      > > any patterns and of course was made quite dizzy. Anyways, one
      thing
      > > that stood out was that if a binary 0 was replaced with a 1 or if
      a
      > > binary 1 was replaced with a 0, then another prime would result.
      In
      > > other words, if we start with a prime, then at least one other
      prime
      > > can be produced by adding or subtracting 2^n, where 2^n is less
      than
      > > the original prime.
      > >
      > > For instance,
      > >
      > > 3-1 = 2; 3+2 = 5 (1 is 2^0)
      >
      > You mention 2 rules.
      > First you say: Only change bit value 0 to 1, or bit value 1 to 0.
      > Then you say: Add or subtract 2^n regardless of the value of that
      bit, e.g.
      > 3+2 = 5 in binary is 11+10=101.
      > Your examples follow the second rule so I do the same.
      >
      > The primes p<50 million setting a new record for most generated
      primes:
      > primes(p).
      > primes(3)=2, primes(7)=3, primes(11)=4, primes(67)=5 , primes(571)
      =6,
      > primes(1487)=7, primes(11831)=8, primes(34369)=9, primes(92639)=10,
      > primes(133319)=11, primes(2233531)=12, primes(18230483)=13,
      > primes(47924959)=14.
      >
      > The first 1: primes(373)=1
      > The first 0's: primes(p)=0 for p=
      > 1973, 3181, 3967, 4889, 8363, 8923, 11437, 12517, 14489, 19583,
      19819, ...
      > There are 42183 0's below 50 million :-(
      >
      > The average number of generated primes from 2 to 50 million is:
      > 11227739/3001134 = 3.741
      > This is twice what Mike Oakes got, not surprising since he followed
      the first
      > rule with half as many candidates.
      >
      > For a related problem where the goal is all primes up to n_max, see
      > www.primepuzzles.net/puzzles/puzz_167.htm
      > Phil Carmody found the impressive 16-tuple:
      > 64606701602327559675+/-2^n is prime for n=1,2,3,4,5,6,7,8.
      > The number itself did not have to be prime in this puzzle.
      >
      > --
      > Jens Kruse Andersen
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