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• A beautifull day, i have started a small collection of primes between 1000 and 10000 digit primes. The primes are all of the form p:=x^2+x+1=x(x+1)+1 I
Message 1 of 18 , Sep 9, 2010
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A beautifull day,

i have started a small collection of primes between 1000 and 10000 digit primes.

The primes are all of the form p:=x^2+x+1=x(x+1)+1

I calculated x of the product of small numbers in order that i could
verify them with pfgw -t. I made first a fermat test and if this one is succesfull, i wrote down the number and checked it later.

The collection is availible under
http://devalco.de 16. Collection of Primes

the collection includes :
1000 digit primes 17514 primes chance for finding
2000 " " 5635 " 1:260
3000 " " 3984 " 1:434
4000 " " 770 " 1:620
5000 " " 385 " 1:810
6000 " " 219 " 1:1050
7000 " " 146 "
8000 " " 237 " 1:1633
9000 " " 238 "
10000 " " 113 " 1:1814

I think the distribution of primes concerning the polynom f(x)=x^2+x+1
is better than the linear distribution
or in other words the chance to find one is better.

I needed 3 month to calculate the numbers on 6 cores.

I think i can improve the chance for finding primes on this polynom
by calculation of x by multiplication of small primes which
can be found on these polynom.

Besides all and only (except 3) primes = 1 mod 6
appear as primes or divisior of f(x)=x^2 +x+1

Nice Greetings from the primes
Bernhard
• ... x^2+x+1 is prime with probability asymptotic to C/log(x^2), where C = prod(p,1-kronecker(-3,p)/(p-1)) with a product over all primes, p. Bernhard has
Message 1 of 18 , Sep 9, 2010
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"bhelmes_1" <bhelmes@...> wrote:

> I think the distribution of primes concerning the polynom
> f(x)=x^2+x+1 is better than the linear distribution

x^2+x+1 is prime with probability asymptotic to C/log(x^2), where
C = prod(p,1-kronecker(-3,p)/(p-1))
with a product over all primes, p.

Bernhard has observed that C > 2.
I shall be somewhat more precise:

C=
2.241465507098582781236668372470874433527110162241\
32536355383092271281135434409630277148141523001173\
60843050218571111781772521331290036121264851241797\
879383961418406006110408676488798831671040955287287...

If you are interested in computing such numbers, see
> High precision computation of Hardy-Littlewood constants
at http://www.math.u-bordeaux1.fr/~cohen/

David
• A beautifull day, there are 32 new 20000 digit primes in the collection :-) http://beablue.selfip.net/devalco/Collection/20000/ (needed time 14 days on 6
Message 1 of 18 , Sep 17, 2010
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A beautifull day,

there are 32 new 20000 digit primes in the collection :-)
http://beablue.selfip.net/devalco/Collection/20000/

(needed time 14 days on 6 cores)

the chance for me to find one is 1 : 2300
i think that is not bad.

how probably is to find the next mersenne prime ?

i improved the algorithm by multipling all small primes mod 6 = 1
< 14000
for x, multiply some random numbers with x,
calculate x(x+1)+1 and test the number

> i have started a small collection of primes between 1000 and 10000 digit primes.
>
> The primes are all of the form p:=x^2+x+1=x(x+1)+1
>
> I calculated x of the product of small numbers in order that i could
> verify them with pfgw -t. I made first a fermat test and if this one is succesfull, i wrote down the number and checked it later.
>
> The collection is availible under
> http://devalco.de 16. Collection of Primes
>
> the collection includes :
> 1000 digit primes 17514 primes chance for finding
> 2000 " " 5635 " 1:260
> 3000 " " 3984 " 1:434
> 4000 " " 770 " 1:620
> 5000 " " 385 " 1:810
> 6000 " " 219 " 1:1050
> 7000 " " 146 "
> 8000 " " 237 " 1:1633
> 9000 " " 238 "
> 10000 " " 113 " 1:1814
20000 " " 32 " 1:2300

Greetings from the primes
Bernhard
• A beautifull day, there are 22 new 30000 digit primes in the collection :-) http://beablue.selfip.net/devalco/Collection/30000/ (needed time 30 days on 6
Message 1 of 18 , Oct 22, 2010
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A beautifull day,

there are 22 new 30000 digit primes in the collection :-)
http://beablue.selfip.net/devalco/Collection/30000/

(needed time 30 days on 6 cores)

the chance for me to find one is approximitly 1 : 3000
i think that is not bad.

how probably is to find the next mersenne prime ?

The collection is availible under
http://devalco.de 16. Collection of Primes

> > the collection includes :
> > 1000 digit primes 17514 primes chance for finding
> > 2000 " " 5635 " 1:260
> > 3000 " " 3984 " 1:434
> > 4000 " " 770 " 1:620
> > 5000 " " 385 " 1:810
> > 6000 " " 219 " 1:1050
> > 7000 " " 146 "
> > 8000 " " 237 " 1:1633
> > 9000 " " 238 "
> > 10000 " " 113 " 1:1814
> 20000 " " 32 " 1:2300
30000 " " 22 " 1:3000

Greetings from the primes
Bernhard
• A beautifull day, there are 15 new 40000 digit primes in the collection :-) http://beablue.selfip.net/devalco/Collection/40000/ (needed time 30 days on 6
Message 1 of 18 , Nov 3, 2010
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A beautifull day,

there are 15 new 40000 digit primes in the collection :-)
http://beablue.selfip.net/devalco/Collection/40000/

(needed time 30 days on 6 cores)

the chance for me to find one is something about 1 : 3600
this result is unsure because i only found 15 primes

Nevertheless the chance to find some primes with 100000 digits
may be around 1:10000

Are there any other statistics about other special primes
about the densitys of primes / the chance to find one ?

I will run my programm for a month for 50000 digit primes
and will try to get some more exact numbers of runtime / test (one fermat test with gmp) and some other values.

if someone is interested to participate in running my c-programm,
i will send it to him

Small explication of the algorithm:
i search for primes on the polynom n^2+n+1 = n(n+1)+1
calculate n by multiplication of all small primes with p-1 | 6 below a limit and some random numbers
only make a fermat test and test the founded candidates by pfgw

>
> how probably is to find the next mersenne prime ?
>
> The collection is availible under
> http://devalco.de 16. Collection of Primes
>
> > > the collection includes :
> > > 1000 digit primes 17514 primes chance for finding
> > > 2000 " " 5635 " 1:260
> > > 3000 " " 3984 " 1:434
> > > 4000 " " 770 " 1:620
> > > 5000 " " 385 " 1:810
> > > 6000 " " 219 " 1:1050
> > > 7000 " " 146 "
> > > 8000 " " 237 " 1:1633
> > > 9000 " " 238 "
> > > 10000 " " 113 " 1:1814
> > 20000 " " 32 " 1:2300
> 30000 " " 22 " 1:3000
40000 " " 15 " 1:3600 ???

Greetings from the primes
Bernhard
• ... Yes. It s very simple, Bernhard, and was long since exposed ... If you sieve any target N to sufficient depth p and and find that N still has no identified
Message 1 of 18 , Nov 4, 2010
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> Are there any other statistics about other special primes
> about the densitys of primes / the chance to find one ?

Yes. It's very simple, Bernhard, and was long since exposed
by Mertens' formula:

> Mertens, F.
> "Ein Beitrag zur analytischen Zahlentheorie."
> J. reine angew. Math. 78, 46-62, 1874.

If you sieve any target N to sufficient depth p and
and find that N still has no identified factor, then
the probability that N is prime is, heuristically:

P = exp(Euler)*log(p)/log(N)

where Euler = 0.577215664901532860606512090082402431...

If you ever succeed in out-performing this heuristic,
on a reliable basis, then you will become very famous.

If you do no better than this heuristic, then you will
be like the rest of us.

If you do worse than this heuristic, then something has

Alles gute

David
• Dear David ... there must be a difference in the distribution of primes in linear progression and the distribution of primes concerning irreducible polynoms
Message 1 of 18 , Nov 19, 2010
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Dear David

> > Are there any other statistics about other special primes
> > about the densitys of primes / the chance to find one ?
>
> Yes. It's very simple, Bernhard, and was long since exposed
> by Mertens' formula:
>
> > Mertens, F.
> > "Ein Beitrag zur analytischen Zahlentheorie."
> > J. reine angew. Math. 78, 46-62, 1874.
>
> If you sieve any target N to sufficient depth p and
> and find that N still has no identified factor, then
> the probability that N is prime is, heuristically:
>
> P = exp(Euler)*log(p)/log(N)
>
> where Euler = 0.577215664901532860606512090082402431...
>

there must be a difference in the distribution of primes
in linear progression and the distribution of primes concerning
irreducible polynoms like p:=n^2+n+1

As far as i see primes of the form of p:=n^2+n+1 appear
more often as the linear distribution.

Please give me a hint if i am wrong or not

Nice Greetings from the primes
Bernhard
• ... Not after deep sieving, as I had explained. We believe that a sieve effectively removes all memory of origin, leaving the Mertens probability that I gave
Message 1 of 18 , Nov 19, 2010
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"bhelmes_1" <bhelmes@...> wrote:

> there must be a difference in the distribution of primes
> in linear progression and the distribution of primes concerning
> irreducible polynoms like p:=n^2+n+1
>
> As far as i see primes of the form of p:=n^2+n+1 appear
> more often as the linear distribution.

Not after deep sieving, as I had explained.
We believe that a sieve effectively removes all memory of
origin, leaving the Mertens probability that I gave you.

No-one has given convincing evidence to challenge my
assertion that the Mertens estimate applies to all samples,
irrespective of what "Prime Form" you chose before sieving.
in a given length of sieving time.

David
• Dear David ... There are 11 50000 digits primes in the database http://beablue.selfip.net/devalco/Collection/50000/ My investigation confirms the Mertens
Message 1 of 18 , Dec 1 1:22 PM
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Dear David

> Not after deep sieving, as I had explained.
> We believe that a sieve effectively removes all memory of
> origin, leaving the Mertens probability that I gave you.
>

There are 11 50000 digits primes in the database
http://beablue.selfip.net/devalco/Collection/50000/

My investigation confirms the Mertens probablity:

AMD 5000+ Ubuntu 8.0.4 g++ 3.4 gmp 5.0.1

Digits Runtime Presieve Calculated Mertens Formula
1000 < 1933 1: 170 1: 171
2000 0.1 sec, < 4243 1: 300 1: 309
3000 0.3 sec, < 6607 1: 434 1: 441
4000 0.6 sec, < 8929 1: 523 1: 568
5000 1,2 sec, < 11257 1: 810 1: 693
6000 1.8 sec, < 13627 1:1050 1: 814
8000 3,6 sec, < 18181 1:1633 1: 1054
10000 6,3 sec, < 22861 1:1814 1: 1289
20000 34,0 sec, < 46141 1:2300 1: 2408
30000 1,30 min < 69379 1:3000 1: 3480
40000 3 min, < 92317 1:3600 1: 4524
50000 6 min < 115549 1:5200 1: 4631
60000 7,35 min, < 138577 1: 6553
80000 15 min, < 184843 1: 8531
100000 27 min, < 230563 1: 10471
200000 132 min, < 461677 1: 19828
1000000 < 2304553 1: 88262
10000000 < 23035273 1:762769

Nice Greetings from the primes
Bernhard
• ... Thanks, Bernhard, for taking the time to check that conjecture. Previously, Mike Oakes had done wonderful work, confirming it to much better statistical
Message 1 of 18 , Dec 1 8:30 PM
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"bhelmes_1" <bhelmes@...> wrote:

> My investigation confirms the Mertens probability

Thanks, Bernhard, for taking the time to check that conjecture.
Previously, Mike Oakes had done wonderful work, confirming it
to much better statistical accuracy.

Regarding your very limited statistics, at 50k digits,
I remark that
> 50000 6 min < 115549 1:5200 1: 4631
is indeed in good accord with Mertens + PNT.
You found only 11 primes and it is pleasing that the ratio
5200/4631 differs from unity by less than 1/sqrt(11).

Alles gute, alles schöne!

David
• A beautiful day, i tried to hunt some 100000 digit primes. i found two primes , the first i could affirm with pfgw t p_100000_a
Message 1 of 18 , Dec 21 2:52 AM
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A beautiful day,

i tried to hunt some 100000 digit primes.

i found two "primes", the first i could affirm with pfgw t p_100000_a
http://beablue.selfip.net/devalco/Collection/100000/

Concerning the second "prime" i did not get a good result
there is the error log file of pfgw
http://beablue.selfip.net/devalco/Collection/100000/pfgw_err.log

Perhaps someone knows how to make a deterministic test for this primes
and perhaps this number can be usefull to improve pfgw.

i made 12000 test to find 2 candidates for a deterministic prime in
20 days on 12 cores, what is not bad, i think

Nice greetings from the primes
Bernhard

http://devalco.de
• ... Try using an up-to-date version of OpenPFGW: http://sourceforge.net/projects/openpfgw/files/ I have set it running with the -t option and so far ...
Message 1 of 18 , Dec 21 3:49 AM
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"bhelmes_1" <bhelmes@...> wrote:

> how to make a deterministic test for this prime
> Expr = 6478306796723098....5899455353125001

Try using an up-to-date version of OpenPFGW:
http://sourceforge.net/projects/openpfgw/files/

I have set it running with the -t option and so far

> Running N-1 test using base 29
...
> N-1: 6478306796723098....5899455353125001 232500/1749078 mro=0

indicates no round-off problem. If this base yields no non-trivial
gcd, then the test should complete in a couple of hours.

David
• ... I remark that this candidate is of the form x^2+x+1 where x has no prime divisor greater than 1524493. Hence the BLS proof should be straightforward. So
Message 1 of 18 , Dec 21 4:43 AM
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> N-1: 6478306796723098....5899455353125001 232500/1749078 mro=0

I remark that this candidate is of the form x^2+x+1
where x has no prime divisor greater than 1524493.
Hence the BLS proof should be straightforward.
So far, N-1 has reached

> N-1: 6478306796723098....5899455353125001 670000/1749078 mro=0

without incident.

David
• ... Indeed it was: PFGW Version 3.4.3.64BIT.20101025.x86_Dev [GWNUM 26.4] [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 29 Calling
Message 1 of 18 , Dec 21 6:37 AM
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> I remark that this candidate is of the form x^2+x+1
> where x has no prime divisor greater than 1524493.
> Hence the BLS proof should be straightforward.

Indeed it was:

PFGW Version 3.4.3.64BIT.20101025.x86_Dev [GWNUM 26.4]
[N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 29
Calling Brillhart-Lehmer-Selfridge with factored part 33.33%
6478306796723098364703559651976954926....
4884247365446132388985899455353125001
is prime! (10116.8919s+0.0854s)

Conclusion: Bernhard's OpenPFGW was way out of date?

David
• Dear David, thank you for your efforts. Is there an interest to submit the two 100000 digit primes for the collection of http://primes.utm.edu/ If yes, give me
Message 1 of 18 , Dec 21 8:49 AM
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Dear David,

Is there an interest to submit the two 100000 digit primes for the collection of http://primes.utm.edu/

If yes, give me please a short link or description, how to submit them

Greetings from the primes
Bernhard
• ... They are too small: http://primes.utm.edu/primes/submit.php ... David
Message 1 of 18 , Dec 21 9:32 AM
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"bhelmes_1" <bhelmes@...> wrote:

> Is there an interest to submit the two 100000 digit
> primes for the collection of http://primes.utm.edu/

They are too small:
http://primes.utm.edu/primes/submit.php
> Currently primes must have 178698 or more digits

David
• Dear David, i did a little investigation concerning the distribution of primes concerning the polynom f(x)=x^2+1
Message 1 of 18 , Oct 28, 2012
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Dear David,

i did a little investigation concerning the distribution of primes
concerning the polynom f(x)=x^2+1

I think it is sensefull to make a presieving of the search array
for searching huge primes and regard the numbers which are divided
by the small primes also.

I have no analytic result how big the improvement is.
Do you know any similar analytic results ?

Nice Greetings from the primes
Bernhard

>
>
>
>
> > Are there any other statistics about other special primes
> > about the densitys of primes / the chance to find one ?
>
> Yes. It's very simple, Bernhard, and was long since exposed
> by Mertens' formula:
>
> > Mertens, F.
> > "Ein Beitrag zur analytischen Zahlentheorie."
> > J. reine angew. Math. 78, 46-62, 1874.
>
> If you sieve any target N to sufficient depth p and
> and find that N still has no identified factor, then
> the probability that N is prime is, heuristically:
>
> P = exp(Euler)*log(p)/log(N)
>
> where Euler = 0.577215664901532860606512090082402431...
>
> If you ever succeed in out-performing this heuristic,
> on a reliable basis, then you will become very famous.
>
> If you do no better than this heuristic, then you will
> be like the rest of us.
>
> If you do worse than this heuristic, then something has
>
> Alles gute
>
> David
>
• ... Since 1923, we have a had a very precise conjecture for the asymptotic density of primes of the form x^2+1. See Shanks review
Message 1 of 18 , Oct 29, 2012
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"bhelmes_1" <bhelmes@...> wrote:

> the distribution of primes
> concerning the polynom f(x)=x^2+1

Since 1923, we have a had a very precise
conjecture for the asymptotic density
of primes of the form x^2+1. See Shanks' review

http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0120203-6/S0025-5718-1960-0120203-6.pdf

of the classic paper by G.H. Hardy and J.E. Littlewood:
"Some problems of 'Partitio numerorum'; III",
Acta Math. 44 (1923) pages 170.

The relevant Hardy-Littlewood constant,
1.3728134... is given, to 9 significant figures,
in Eq(3) of Shanks' paper.

More digits are easily obtainable from the methods in
"High precision computation of Hardy-Littlewood constants"
by Henri Cohen, available as a .dvi file from
http://www.math.u-bordeaux1.fr/~hecohen/

David
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