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• ## There are Infinite Number that cannot be Proven !

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• About the issue, we can look at it this way : infinity can not be proven to be prime or not, niether can infinity -1 and infinity -2 .... Therefore no
Message 1 of 10 , Feb 12, 2001
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About the issue, we can look at it this way :
infinity can not be proven to be prime or not,
niether can
infinity -1 and
infinity -2 ....
Therefore no matter what computing power you have or what
mathematical solution humans come up with, there are infinit numbers
can not be proved prime or not !

Habib
• ... But..... infinity, infinity-1, infinity-2, infinty-k (where k is a -finite- natural) are not numbers [Non-text portions of this message have been removed]
Message 1 of 10 , Feb 12, 2001
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> About the issue, we can look at it this way :
> infinity can not be proven to be prime or not, niether can
> infinity -1 and
> infinity -2 ....
> Therefore no matter what computing power you have or what
> mathematical solution humans come up with, there are infinit numbers
> can not be proved prime or not !

But.....
infinity, infinity-1, infinity-2, infinty-k (where k is a -finite- natural) are not numbers

[Non-text portions of this message have been removed]
• ... I think inf-1 must be prime as infinity contains all numbers as factors. If so, inf-2 = 2*prime because out of any two consecutive even numbers, one will
Message 1 of 10 , Feb 12, 2001
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--- habibnajafi@... wrote:
> About the issue, we can look at it this way :
> infinity can not be proven to be prime or not,
> niether can
> infinity -1 and
> infinity -2 ....

I think inf-1 must be prime as infinity contains
all numbers as factors.
If so, inf-2 = 2*prime because out of any two
consecutive even numbers, one will be 2*odd and the other
will be (2^a)*odd, a>1.
For infinity, a is very large,
thus inf-2 is of form 2*odd, but
two consecutive even numbers cannot
share an odd factor, therefore (inf-2)/2
is prime.
Suggests primes of the form (2^a)*N#-1
could be good pickings:)

Dick Boland

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• At 03:07 PM 2/12/2001 -0800, Dick Boland wrote: I think inf-1 must be prime as infinity contains ... There s no such integer as infinity-1.
Message 1 of 10 , Feb 12, 2001
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At 03:07 PM 2/12/2001 -0800, Dick Boland wrote:
I think inf-1 must be prime as infinity contains
>all numbers as factors.

There's no such integer as infinity-1.

+--------------------------------------------------------+
| Jud McCranie |
| |
| 137*2^261147+1 is prime! (78,616 digits, 5/2/00) |
+--------------------------------------------------------+
• In message , Dick Boland writes ... So, is it true that (p1*p2*p3...)-1 is always
Message 1 of 10 , Feb 12, 2001
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In message <20010212230732.12254.qmail@...>, Dick
Boland <richard042@...> writes
>I think inf-1 must be prime as infinity contains
>all numbers as factors.

So, is it true that (p1*p2*p3...)-1 is always prime?
--
Ben
• ... 2*3*5*7=210 210-1=209=11*19 It /is/ true that the prime factors of such a value will never include the factors used in producing it - this is the basis of
Message 1 of 10 , Feb 12, 2001
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On Tue, 13 Feb 2001 02:14:31 +0000, Ben Newsam wrote:

>So, is it true that (p1*p2*p3...)-1 is always prime?

2*3*5*7=210
210-1=209=11*19

It /is/ true that the prime factors of such a value will never include
the factors used in producing it - this is the basis of Euclid's proof
of the infinitude of primes.

Nathan
• ... Which prime times 2 equals infinity-2? ... So infinity is even! +--------------------------------------------------------+ ...
Message 1 of 10 , Feb 12, 2001
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At 03:07 PM 2/12/2001 -0800, Dick Boland wrote:

> If so, inf-2 = 2*prime because out of any two
>consecutive even numbers, one will be 2*odd and the other

Which prime times 2 equals infinity-2?

...

> thus inf-2 is of form 2*odd,

So infinity is even!

+--------------------------------------------------------+
| Jud McCranie |
| |
| 137*2^261147+1 is prime! (78,616 digits, 5/2/00) |
+--------------------------------------------------------+
• ... No, obviously not. Just intimating that the form 2^a*N#-1 has a higher probability of being prime over a random number and the lower the a, the fewer
Message 1 of 10 , Feb 12, 2001
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--- Ben Newsam <primes@...> wrote:

> So, is it true that (p1*p2*p3...)-1 is always prime?

No, obviously not. Just intimating that
the form 2^a*N#-1 has a higher
probability of being prime over a random
number and the lower the a, the fewer
primes between the highest prime
in your primorial and the primorial itself.
of the difference quickly goes to zero.
And I wonder, if 2^a*N#-1 is prime for
some a,N where a is large,
Is (2^a-2^b)*N#-1 more likely to be
prime than for some a,N where
2^a*N#-1 is not prime?

Here's an interesting challenge.

Who can find the largest "tri-morial"
of 3 primes such that
2^b*N#-1 is prime
(2^a-2^b)*N#-1 is prime and
2^a*N#-1 is prime (largest of 3)
(a-b)>1
b will have to be sufficently
large to ensure provability.

Does it happen at all?
Does it happen infinitely?
Does it happen more than once for a given N?
If (2^a+2^b)*N#-1 were also prime, that would be neat.

-Dick

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• ... I did do a search of these numbers (and found one of his neat numbers.) The method I used shows part of what can be done with the ABC2 format. I simply
Message 1 of 10 , Feb 14, 2001
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At 09:51 AM 2/14/01 -0000, you wrote:
>
>I think this could be a useful education in how to use the ABC file format for PFGW.
>
>I believe that it can be done in 2 passed, one to generate the 2^a.N#-1 terms, and then an ABC2 file to iterate over the primes in that set twice, looking at the differences.

I did do a search of these numbers (and found one of his
"neat" numbers.) The method I used shows part of what can
be done with the ABC2 format. I simply created a file

ABC2 2^\$a*\$b-1
a: from 1 to 1400
b: primes from 149 to 181

I ran this, and the resultant file contained all of the
2^b*N#-1 and 2^a*N-1 possibles. I then created separate
ABC2 files for each N. Within N=191 up to k=300 (used
for this example, since it is short enough to show for
this email), these primes are found:

2^33*191#-1
2^46*191#-1
2^58*191#-1
2^59*191#-1
2^73*191#-1
2^74*191#-1
2^87*191#-1
2^108*191#-1
2^127*191#-1
2^140*191#-1
2^172*191#-1
2^191*191#-1
2^212*191#-1
2^298*191#-1

So now I simply create a ABC2 file

ABC2 (2^\$a-2^\$b)*191#-1
a: in { 33 46 58 59 73 74 87 108 127 140 172 191 212 298 }
b: in { 33 46 58 59 73 74 87 108 127 140 172 191 212 }

and then run this. The pfgw.log file contains:

(2^74-2^33)*191#-1
(2^59-2^58)*191#-1
(2^298-2^59)*191#-1
(2^74-2^73)*191#-1
(2^172-2^87)*191#-1

which all turn out to have a>b (positive numbers). The
pfgw.log file has to be "edited, since there will (can)
be number contained within it that have b>a (pfgw will
run the test on -N when N is negative). If the above
file contained (2^73-2^140)*191#-1 then this would be
the result of the negation of negative number being
prime.

All in all, pfgw and the ABC2 format make this search
pretty simple. I have replied to Dick about my "findings".
Numbers of this type are common enough (at least in the
300 digit range I was looking at. They are surely infinite
in nature, and probably infinite for most if not all prime
N.

Jim.

>
>There's the extra bonus of the "+ being neat" providing an example of the logical operators too.
>
>Isn't that almost every feature we have presently?
>
>Phil
>
>On Mon, 12 February 2001, Dick Boland wrote:
>> Here's an interesting challenge.
>>
>> Who can find the largest "tri-morial"
>> of 3 primes such that
>> 2^b*N#-1 is prime
>> (2^a-2^b)*N#-1 is prime and
>> 2^a*N#-1 is prime (largest of 3)
>> (a-b)>1
>> b will have to be sufficently
>> large to ensure provability.
>>
>> Does it happen at all?
>> Does it happen infinitely?
>> Does it happen more than once for a given N?
>> If (2^a+2^b)*N#-1 were also prime, that would be neat.
>>
>> -Dick
• Hi, This search definitely points out the strengths and weaknesses of the ABC2 mode. I first thought that it might be done using the logical operators, and it
Message 1 of 10 , Feb 14, 2001
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Hi,

This search definitely points out the strengths and weaknesses of the ABC2
mode. I first thought that it might be done using the logical operators,
and it can, but only if you are prepared to watch PFGW test the same numbers
again and again hundreds of times. It would be nice if a specific variable
could be looped only if the first expression was prime or something like
that, I'll think about that at some time.

However, it also shows how "in" can be used to great effect, and using the
method describe below I found,
2^252*401#-1, (2^849-2^525)*401#-1 and 2^849*401#-1 are all prime, and also
2^307*409#-1, (2^507-2^307)*409#-1 and 2^507*409#-1 are prime.

Michael.

----- Original Message -----
From: "Jim Fougeron" <jfoug@...>
Sent: Wednesday, February 14, 2001 1:32 PM
Subject: Re: [PrimeNumbers] There are Infinitely many that can be Proven !

> At 09:51 AM 2/14/01 -0000, you wrote:
> >
> >I think this could be a useful education in how to use the ABC file
format for PFGW.
> >
> >I believe that it can be done in 2 passed, one to generate the 2^a.N#-1
terms, and then an ABC2 file to iterate over the primes in that set twice,
looking at the differences.
>
> I did do a search of these numbers (and found one of his
> "neat" numbers.) The method I used shows part of what can
> be done with the ABC2 format. I simply created a file
>
> ABC2 2^\$a*\$b-1
> a: from 1 to 1400
> b: primes from 149 to 181
>
> I ran this, and the resultant file contained all of the
> 2^b*N#-1 and 2^a*N-1 possibles. I then created separate
> ABC2 files for each N. Within N=191 up to k=300 (used
> for this example, since it is short enough to show for
> this email), these primes are found:
>
> 2^33*191#-1
> 2^46*191#-1
> 2^58*191#-1
> 2^59*191#-1
> 2^73*191#-1
> 2^74*191#-1
> 2^87*191#-1
> 2^108*191#-1
> 2^127*191#-1
> 2^140*191#-1
> 2^172*191#-1
> 2^191*191#-1
> 2^212*191#-1
> 2^298*191#-1
>
> So now I simply create a ABC2 file
>
> ABC2 (2^\$a-2^\$b)*191#-1
> a: in { 33 46 58 59 73 74 87 108 127 140 172 191 212 298 }
> b: in { 33 46 58 59 73 74 87 108 127 140 172 191 212 }
>
> and then run this. The pfgw.log file contains:
>
> (2^74-2^33)*191#-1
> (2^59-2^58)*191#-1
> (2^298-2^59)*191#-1
> (2^74-2^73)*191#-1
> (2^172-2^87)*191#-1
>
> which all turn out to have a>b (positive numbers). The
> pfgw.log file has to be "edited, since there will (can)
> be number contained within it that have b>a (pfgw will
> run the test on -N when N is negative). If the above
> file contained (2^73-2^140)*191#-1 then this would be
> the result of the negation of negative number being
> prime.
>
> All in all, pfgw and the ABC2 format make this search
> pretty simple. I have replied to Dick about my "findings".
> Numbers of this type are common enough (at least in the
> 300 digit range I was looking at. They are surely infinite
> in nature, and probably infinite for most if not all prime
> N.
>
> Jim.
>
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