## Re: [PrimeNumbers] Infinite Number of Twin Primes

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• Perhaps a computer search would be better for you. Or, see the link http://primes.utm.edu/lists/small/1ktwins.txt (7!+59, 7!+61) = (5099, 5101) is a prime
Message 1 of 13 , May 5 5:36 PM
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Perhaps a computer search would be better for you.

(7!+59, 7!+61) = (5099, 5101) is a prime pair.

Milton L. Brown

> [Original Message]
> From: D�cio Luiz Gazzoni Filho <decio@...>
> Date: 5/5/2005 4:14:35 PM
> Subject: Re: [PrimeNumbers] Infinite Number of Twin Primes
>
> On Thursday 05 May 2005 19:49, you wrote:
> > At 04:54 PM 5/5/2005, Milton Brown wrote:
> > >This is the pattern: Start at the next factorial and add prior pairs of
> > >twin primes,
> > >until you obtain a pair of twin primes for that factorial.
> >
> > How do you know that will work (that you will always obtain a pair of
twins
> > for that factorial)?
>
> The answer is, he doesn't, because it won't work. This one didn't even
need a
> computer search; it fell to a search by hand. I would appreciate a third
> check (I've already double-checked locally).
>
> Start from (3,5).
>
> 2!+3, 2!+5 adds (5,7). Values found until now: (3,5), (5,7).
>
> Values found until now: (3,5), (5,7), (11,13), (17,19).
>
> Values found until now: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43).
>
> (269,271).
> Values found until now: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43),
> (137,139), (149,151), (269,271).
>
> Values found until now: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43),
> (137,139), (149,151), (269,271), (857,859).
>
> Finally, 7! doesn't produce any values. And thus Milton's conjecture is
> demolished, as about anything that he brainfarts on this list.
>
> D�cio
>
>
> [Non-text portions of this message have been removed]
>
>
>
>
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> The Prime Pages : http://www.primepages.org/
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• Even a computer search does not help you! (3629027, 3629029) = (10!+227, 10!+229) Unless n=10 means something else to you. (Maybe its not the computer after
Message 2 of 13 , May 5 9:04 PM
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(3629027, 3629029) = (10!+227, 10!+229)

Unless n=10 means something else to you.

(Maybe its not the computer after all)

Milton L. Brown

> [Original Message]
> From: D�cio Luiz Gazzoni Filho <decio@...>
> Date: 5/5/2005 8:06:20 PM
> Subject: Re: [PrimeNumbers] Infinite Number of Twin Primes
>
> On Thursday 05 May 2005 20:13, you wrote:
> > Finally, 7! doesn't produce any values. And thus Milton's conjecture is
> > demolished, as about anything that he brainfarts on this list.
>
> Moreover, a PARI/GP search plus a clever shell script using PFGW (guess I
> should learn PFGW's scripting language instead, but...) confirms my
previous
> findings, and adds failures at n! for n = 10, 12-14, 16, 17, 22-64,
66-70,
> 72-135, 137-361, 363-683,685-900; that's as far as I'm willing to take
this
> search. I have reason to believe that only finitely many twin prime pairs
are
> generated by this process.
>
> Additionally, in the course of running this code, I discovered a bug in
> PARI/GP 2.2.9:
>
> ? isprime(42542905343533366778773944705953203289361426945380795183689)
> *** isprime: impossible inverse modulo: Mod(0,
> 42542905343533366778773944705953203289361426945380795183689).
>
> Ironically, Milton's work was not 100% in vain.
>
> D�cio
>
> PS: in principle one can actually check my conjecture that the sequence
is
> finite: if p, p+2 is the largest twin prime pair generated, then test all
n's
> up to n = p. Obviously p! shares a factor with all twin primes in the
set,
> and so does n! for all n > p. Thus, no new twin prime pairs can be
generated
> from then on. Of course, exhausting the range up to n! when n is in the
> thousands of digits isn't exactly practical.
>
>
> [Non-text portions of this message have been removed]
>
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> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
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• How many fact-twins are there ? A pair of fact-twins are twins primes (ft, ft+2) if exists n such as ft = n! + p, where p and p+2 are twin primes; The
Message 3 of 13 , May 6 9:32 AM
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How many fact-twins are there ?

A pair of fact-twins are twins primes (ft, ft+2) if exists n such as
ft = n! + p, where p and p+2 are twin primes;

The following table count the number of fact-twins in intervals of length
10^9 starting at
0, up to 10^18.

pi = number of primes in the interval [from, from+10^9]
twins = number of twins
fact-twins : number of fact-twins
% = fact-twins * 100 / twins

from pi twins fact-twins
%
1 50847534 3424506 999836 29,20%
10^9 47374753 2963535 815419 27,51%
10^10 43336106 2477174 674611 27,23%
10^11 39475591 2055627 547106 26,62%
10^12 36190991 1730012 386360 22,33%
10^13 33405006 1473196 318563 21,62%
10^14 31019409 1270499 265221 20,88%
10^15 28946421 1105560 225366 20,38%
10^16 27153205 972510 194566 20,01%
10^17 25549226 861742 153809 17,85%
10^18 24127085 769103 135977 17,68%
• ... the point is that you gave this procedure for producing more and more twin primes from factorials and the previously found twin primes. Other than a
Message 4 of 13 , May 6 10:06 AM
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At 03:09 AM 5/6/2005, Milton Brown wrote:
>
> (479001791, 479001793) = (12!+191, 12!+193)
>
>And, again you said 12 didn't work!

the point is that you gave this procedure for producing more and more twin
primes from factorials and the previously found twin primes. Other than a
finite number of twin primes as seeds and the ones previously generated by
procedure has to bootstrap itself and keep going forever. IIRC, he showed
that 191 and 193 can't be generated that way, so you can't use them for 12!.
• Jud s point is well taken. Milton s original offering was that twin primes would be generated from only previously generated twin primes. I took the liberty to
Message 5 of 13 , May 6 10:43 AM
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Jud's point is well taken. Milton's original offering was that twin
primes would be generated from only previously generated twin primes.

I took the liberty to modify Milton's original offering to this:

Starting with n=3 and (5,7) as the first twin prime pair, if we add
*or subtract* a (previously generated) twin prime pair from n! to
generate more twin prime pairs, how far can n go before there is a
failure to produce twins? Answer : n=22.

But if instead of n! we use p#, then we can get as high as p = 61
before there is a failure to produce. (61 is the only prime to fail
before 103. After that failures become more common, as one might
expect.)

Mark

PS to certain others: Where is the love? How can there ever be peace
in the world if we respect our own knowledge or sensibilities or
tradition over our fellow human beings?
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