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