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

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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
Message 1 of 3 , Jun 28, 2003
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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