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 Prime numbers and primality testing

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symmetrical primes
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 Hi all,
Six primes are symmetrically arranged around the number 12:
5, 7, 11, (12) 13, 17, 19.
Six primes are also symmtrically arranged around the number 15:
7, 11, 13, (15) 17, 19, 23.
Ten primes are symmetrically arranged around the number 30:
13, 17, 19, 23, 29, (30) 31, 37, 41, 43, 47.
Twelve primes are symmetrically arranged around the number 165:
137, 139, 149, 151, 157, 163, (165) 167, 173, 179, 181, 191, 193.
I looked up to 100,000 primes there are six cases of twelve primes
symmetrically arranged around a central number. Found nothing higher.
When is the first case of a symmetrical grouping of more than twelve
primes?
Mark  First case of 4 primes:
5 7 (9) 11 13
First case of 6 primes:
7 11 13 (15) 17 19 23
First case of 8 primes:
17 19 23 29 (30) 31 37 41 43
First case of 10 primes:
139 149 151 157 163 (165) 167 173 179 181 191
First case of 12 primes:
55787 55793 55799 55807 55813 55817 (55818) 55819 55823 55829 55837
55843 55849
First case of 14 primes:
8021749 8021753 8021759 8021771 8021789 8021791 8021801 (8021811)
8021821 8021831 8021833 8021851 8021863 8021869 8021873
First case of 16 primes:
1071065111 1071065123 1071065129 1071065137 1071065141 1071065153
1071065167 1071065179 (1071065190) 1071065201 1071065213 1071065227
1071065239 1071065243 1071065251 1071065257 1071065269
First case of 18 primes:
1613902553 1613902561 1613902567 1613902573 1613902601 1613902621
1613902627 1613902643 1613902649 (1613902650) 1613902651 1613902657
1613902673 1613902679 1613902699 1613902727 1613902733 1613902739
1613902747
I can easily do up 2^6459 for you as well

Alan McFarlane
Mark Underwood wrote:> Hi all,
>
> Six primes are symmetrically arranged around the number 12:
> 5, 7, 11, (12) 13, 17, 19.
>
> Six primes are also symmtrically arranged around the number 15:
> 7, 11, 13, (15) 17, 19, 23.
>
> Ten primes are symmetrically arranged around the number 30:
> 13, 17, 19, 23, 29, (30) 31, 37, 41, 43, 47.
>
> Twelve primes are symmetrically arranged around the number 165:
> 137, 139, 149, 151, 157, 163, (165) 167, 173, 179, 181, 191, 193.
>
> I looked up to 100,000 primes there are six cases of twelve primes
> symmetrically arranged around a central number. Found nothing higher.
> When is the first case of a symmetrical grouping of more than twelve
> primes?
>
> Mark
>
>
>
>
>
>
>
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>
>
>  In an email dated Sun, 12 3 2006 4:43:42 am GMT, "Mark Underwood" <mark.underwood@...> writes:
>Six primes are symmetrically arranged around the number 12:
A pretty idea, Mark!
>5, 7, 11, (12) 13, 17, 19.
>
>Six primes are also symmtrically arranged around the number 15:
>7, 11, 13, (15) 17, 19, 23.
>
>Ten primes are symmetrically arranged around the number 30:
>13, 17, 19, 23, 29, (30) 31, 37, 41, 43, 47.
>
>Twelve primes are symmetrically arranged around the number 165:
>137, 139, 149, 151, 157, 163, (165) 167, 173, 179, 181, 191, 193.
>
>I looked up to 100,000 primes there are six cases of twelve primes
>symmetrically arranged around a central number. Found nothing higher.
>When is the first case of a symmetrical grouping of more than twelve
>primes?
>
For primes up to 2*10^9, here are the first occurrences of your patterns, as output by my Pascal program:
count=2:
3, (4) 5,
count=4:
5, 7, (9) 11, 13,
count=6:
5, 7, 11, (12) 13, 17, 19,
count=10:
13, 17, 19, 23, 29, (30) 31, 37, 41, 43, 47,
count=12:
137, 139, 149, 151, 157, 163, (165) 167, 173, 179, 181, 191, 193,
count=14:
8021749, 8021753, 8021759, 8021771, 8021789, 8021791, 8021801, (8021811) 8021821, 8021831, 8021833, 8021851, 8021863, 8021869, 8021873,
count=16:
1071065111, 1071065123, 1071065129, 1071065137, 1071065141, 1071065153, 1071065167, 1071065179, (1071065190) 1071065201, 1071065213, 1071065227, 1071065239, 1071065243, 1071065251, 1071065257, 1071065269,
count=18:
1613902553, 1613902561, 1613902567, 1613902573, 1613902601, 1613902621, 1613902627, 1613902643, 1613902649, (1613902650) 1613902651, 1613902657, 1613902673, 1613902679, 1613902699, 1613902727, 1613902733, 1613902739, 1613902747,
[At 2.08GHz, 3 mins to compute all the primes, 2 secs to find all the patterns.]
Mike Oakes  From: "Mark Underwood" <mark.underwood@...>
> Hi all,
A great puzzle Mark!
>
> Six primes are symmetrically arranged around the number 12:
> 5, 7, 11, (12) 13, 17, 19.
>
> Six primes are also symmtrically arranged around the number 15:
> 7, 11, 13, (15) 17, 19, 23.
>
> Ten primes are symmetrically arranged around the number 30:
> 13, 17, 19, 23, 29, (30) 31, 37, 41, 43, 47.
>
> Twelve primes are symmetrically arranged around the number 165:
> 137, 139, 149, 151, 157, 163, (165) 167, 173, 179, 181, 191, 193.
>
> I looked up to 100,000 primes there are six cases of twelve primes
> symmetrically arranged around a central number. Found nothing higher.
> When is the first case of a symmetrical grouping of more than twelve
> primes?
Shame I misread it  I was trying to stick a prime in the middle...
Some examples:
6 primes symmetrically arranged around a central prime
683747 683759 683777 683783 683789 683807 683819
. +12 +18 +6 +6 +18 +12
8 primes symmetrically arranged around a central prime
98303867 98303873 98303897 98303903 98303927 98303951 98303957 98303981
98303987
. +6 +24 +6 +24 +24 +6 +24 +6
10 primes symmetrically arranged around a central prime
60335249851 +6
60335249857 +12
60335249869 +12
60335249881 +60
60335249941 +18
60335249959 +18
60335249977 +60
60335250037 +12
60335250049 +12
60335250061 +6
60335250067
12  well, that's a job for Jens :)
Quiz question:
Why are my minimal examples so much larger than Mark's examples?
I have the answer. It's quite elementary, but still took 30 seconds of head
scratching before I worked it out. Award yourself a Carmodyapproved pat on the
back if you work it out more quickly.
Yes, I appreciate this does not answer your original question :P
Phil
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http://mail.yahoo.com   In primenumbers@yahoogroups.com, Alan McFarlane
<alan.mcfarlane@...> wrote:> First case of 18 primes:
Easily? Do you realise how big 2^64 is?
> 1613902553 1613902561 1613902567 1613902573 1613902601 1613902621
> 1613902627 1613902643 1613902649 (1613902650) 1613902651 1613902657
> 1613902673 1613902679 1613902699 1613902727 1613902733 1613902739
> 1613902747
>
>
> I can easily do up 2^6459 for you as well
The above length18 case may only be 4 GHz seconds of computation
using a naive algorithm, and going 2^33 times further would imply
about 2GHz millennia.
Phil  Hmm, I noticed that...
I have, however, successfully completed an exhaustive search up to 10^12
and not found any occurences of 20 primes in symetrical sequence.
I'll keep it running for a while, but I may have to resort to running it
on my farm for a week or so :)
thefatphil wrote:>  In primenumbers@yahoogroups.com, Alan McFarlane
> <alan.mcfarlane@...> wrote:
>> First case of 18 primes:
>> 1613902553 1613902561 1613902567 1613902573 1613902601 1613902621
>> 1613902627 1613902643 1613902649 (1613902650) 1613902651 1613902657
>> 1613902673 1613902679 1613902699 1613902727 1613902733 1613902739
>> 1613902747
>>
>>
>> I can easily do up 2^6459 for you as well
>
> Easily? Do you realise how big 2^64 is?
>
> The above length18 case may only be 4 GHz seconds of computation
> using a naive algorithm, and going 2^33 times further would imply
> about 2GHz millennia.
>
> Phil
>
>
>
>
>
>
> Unsubscribe by an email to: primenumbersunsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
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> Yahoo! Groups Links
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>
>
>
>
>
>   Alan McFarlane <alan.mcfarlane@...> wrote:
> Hmm, I noticed that...
Confirmed. (2^40, rather than 10^12)
>
> I have, however, successfully completed an exhaustive search up to 10^12
> and not found any occurences of 20 primes in symetrical sequence.
Quoth a 2GHz Athlon (sharing with a nice PIES process):
bash3.1$ time ./symprimeven
real 101m15.817s
user 84m27.830s
sys 0m2.550s
> I'll keep it running for a while, but I may have to resort to running it
I think it is worth trying to find such a cluster, and also that it's not worth
> on my farm for a week or so :)
competing, and definitely worth cooperating. So you have dibbs on the task (if
you have a farm, that only makes sense  I only have a herb garden!). If my
code's faster than yours, you can have it freely. However, as I hinted, my
code's not very clever.
I was thinking of a branchless FSMlike approach optimisation, but decided to
not bother in the end. However, if you want to try to work on optimising the
task, I can join you in a brainstorm.
Phil
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http://mail.yahoo.com  I'm just in the process of writing an optimized version (ANSI C to start
with), but in the interim, why don't we allocate blocks of data to work on.
Say a block size of 2^40? Seems a reasonable size to me  it's fairly
easy for most machines to do, but is still meaty enough to be interesting.
Block 1 : (0 * 2^40) .. (1 * 2^40)  completed (18 digits found) [AM/PC]
Block 2 : (1 * 2^40) .. (2 * 2^40)
Block 3 : (2 * 2^40) .. (3 * 2^40)
Block 4 : (3 * 2^40) .. (4 * 2^40)
Block 5 : (4 * 2^40) .. (5 * 2^40)
...
etc
If you agree with this, reserve a block or two, I'll do the same, and
see just where we can get to.
BTW, we will need to have a small overlap just in case a sequence is on
a block boundary. It might be an idea to start, say, 100 primes before
the block and end 100 primes after.

Alan
Phil Carmody wrote:>  Alan McFarlane <alan.mcfarlane@...> wrote:
>> Hmm, I noticed that...
>>
>> I have, however, successfully completed an exhaustive search up to 10^12
>> and not found any occurences of 20 primes in symetrical sequence.
>
> Confirmed. (2^40, rather than 10^12)
>
> Quoth a 2GHz Athlon (sharing with a nice PIES process):
>
> bash3.1$ time ./symprimeven
>
> real 101m15.817s
> user 84m27.830s
> sys 0m2.550s
>
>> I'll keep it running for a while, but I may have to resort to running it
>> on my farm for a week or so :)
>
> I think it is worth trying to find such a cluster, and also that it's not worth
> competing, and definitely worth cooperating. So you have dibbs on the task (if
> you have a farm, that only makes sense  I only have a herb garden!). If my
> code's faster than yours, you can have it freely. However, as I hinted, my
> code's not very clever.
>
> I was thinking of a branchless FSMlike approach optimisation, but decided to
> not bother in the end. However, if you want to try to work on optimising the
> task, I can join you in a brainstorm.
>
> Phil
>
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 Alan McFarlane <alan.mcfarlane@...> wrote:> I'm just in the process of writing an optimized version (ANSI C to start
If your C is not fast enough and you're tempted to go into assembler, then
you're not writing fast enough C! (Not always true, of course, but I am
notoriously proC.)
> with), but in the interim, why don't we allocate blocks of data to work on.
That does make some sense. I'll look at optimising my code too, and decide how
> Say a block size of 2^40? Seems a reasonable size to me  it's fairly
> easy for most machines to do, but is still meaty enough to be interesting.
much I want to bite off. I'd normally have my machines running PRPing for Les
GeneFermiers, but can certainly put aside a few hours. Maybe more on a "spare"
ancient machine (a crime against primality!) that I've not powered up for a
while.
> If you agree with this, reserve a block or two, I'll do the same, and
The more efficient code should be run. I'll put another 168 GHz minutes
> see just where we can get to.
onto a chunk:
Block 1 : (0 * 2^40) .. (1 * 2^40)  completed (18 digits found) [AM/PC]
Block 2 : (1 * 2^40) .. (2 * 2^40)  reserved PC
Block 3 : (2 * 2^40) .. (3 * 2^40)
Block 4 : (3 * 2^40) .. (4 * 2^40)
Block 5 : (4 * 2^40) .. (5 * 2^40)
If you tell me my code's faster, I'll grab a few more. If you tell me yours is
faster, I'll only look at my new routine, and put my machine back onto LG.
> BTW, we will need to have a small overlap just in case a sequence is on
Yup, that makes sense. Or at least 20 primes does.
> a block boundary. It might be an idea to start, say, 100 primes before
> the block and end 100 primes after.
I'll do 0x<N>fffffff000 to 0x<N+2>0000000fff .
Phil
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http://mail.yahoo.com   Alan McFarlane <alan.mcfarlane@...> wrote:
> I'll test my code on Block #3
Looks like you stopped to soon first time  the whole block took 2hrs on the
>
> Block 1 : (0 * 240) .. (1 * 240)  completed (18 digits found) [AM/PC]
> Block 2 : (1 * 240) .. (2 * 240)  reserved PC
> Block 3 : (2 * 240) .. (3 * 240)  reserved AM
2GHz Athlon.
1797595815167
1797595815157 +10
1797595815133 +24
1797595815109 +24
1797595815091 +18
1797595815089 +2
1797595815079 +10
1797595815053 +26
1797595815019 +34
1797595815017 +2
1797595815013 +4
1797595815011 +2
1797595814977 +34
1797595814951 +26
1797595814941 +10
1797595814939 +2
1797595814921 +18
1797595814897 +24
1797595814873 +24
1797595814863 +10
Phil
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http://mail.yahoo.com  Good one Phil, well spotted!
I admit to making a couple of minor blunders in my code, but I've
rewritten it now  hopefully it works.
I'm working on block #3, current results as follows:
C:\Documents and Settings\Alan\My Documents\Work in Progress\ssop>ssop 3
[20060312 22:22:57] Initializing
[20060312 22:22:57] Processing block #3 (21990232555523298534948863)
[20060312 22:22:57] Found a sequence of 2 primes starting at
2199023255579 with offsets from 2199023255598 of plus and minus
[20060312 22:22:57] 19
[20060312 22:22:57] Found a sequence of 4 primes starting at
2199023256557 with offsets from 2199023256585 of plus and minus
[20060312 22:22:57] 16 28
[20060312 22:22:57] Found a sequence of 6 primes starting at
2199023258429 with offsets from 2199023258454 of plus and minus
[20060312 22:22:57] 5 23 25
[20060312 22:22:57] Found a sequence of 8 primes starting at
2199023370119 with offsets from 2199023370141 of plus and minus
[20060312 22:22:57] 2 8 20 22
[20060312 22:22:58] Found a sequence of 10 primes starting at
2199027144263 with offsets from 2199027144381 of plus and minus
[20060312 22:22:58] 62 70 82 92 118
[20060312 22:23:06] Found a sequence of 12 primes starting at
2199128142049 with offsets from 2199128142126 of plus and minus
[20060312 22:23:06] 17 35 53 67 73 77
[20060312 22:26:31] Found a sequence of 14 primes starting at
2201711786839 with offsets from 2201711786988 of plus and minus
[20060312 22:26:31] 5 41 71 95 125 145 149
[20060312 22:28:06] Found a sequence of 16 primes starting at
2202939530537 with offsets from 2202939530610 of plus and minus
[20060312 22:28:06] 1 41 43 47 53 59 67 73
[20060312 23:24:00] ...
I don't know when it will finish, as I'm testing for sequences up to 80
in length, but hopefully it should be in another couple of hours or so.
This is a slow machine  AMD Athlon 1.25 GHz, but I'm not running
anything else too heavy on it. (ssop is getting around 90% of processor
time allocated to it).
Phil Carmody wrote:>  Alan McFarlane <alan.mcfarlane@...> wrote:
>> I'll test my code on Block #3
>>
>> Block 1 : (0 * 240) .. (1 * 240)  completed (18 digits found) [AM/PC]
>> Block 2 : (1 * 240) .. (2 * 240)  reserved PC
>> Block 3 : (2 * 240) .. (3 * 240)  reserved AM
>
> Looks like you stopped to soon first time  the whole block took 2hrs on the
> 2GHz Athlon.
>
>
> 1797595815167
> 1797595815157 +10
> 1797595815133 +24
> 1797595815109 +24
> 1797595815091 +18
> 1797595815089 +2
> 1797595815079 +10
> 1797595815053 +26
> 1797595815019 +34
> 1797595815017 +2
> 1797595815013 +4
> 1797595815011 +2
> 1797595814977 +34
> 1797595814951 +26
> 1797595814941 +10
> 1797595814939 +2
> 1797595814921 +18
> 1797595814897 +24
> 1797595814873 +24
> 1797595814863 +10
>
> Phil
>
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>  Great going you guys, a symmetrical 20 cluster is found. Phil, I
noticed that your central number  1797595815015  is very factor
rich (containing all primes from 3 to 17).
I did some head scratching as well as to why first occurances of
cluster sizes are so much larger with a prime as the centre number.
Although it me much longer than 30 seconds to get it, I still gave
myself a pat on the back. :)
Another puzzle is on the way.
Mark
 In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...>
wrote:>
[AM/PC]
>  Alan McFarlane <alan.mcfarlane@...> wrote:
> > I'll test my code on Block #3
> >
> > Block 1 : (0 * 240) .. (1 * 240)  completed (18 digits found)
> > Block 2 : (1 * 240) .. (2 * 240)  reserved PC
2hrs on the
> > Block 3 : (2 * 240) .. (3 * 240)  reserved AM
>
> Looks like you stopped to soon first time  the whole block took
> 2GHz Athlon.
>
>
> 1797595815167
> 1797595815157 +10
> 1797595815133 +24
> 1797595815109 +24
> 1797595815091 +18
> 1797595815089 +2
> 1797595815079 +10
> 1797595815053 +26
> 1797595815019 +34
> 1797595815017 +2
> 1797595815013 +4
> 1797595815011 +2
> 1797595814977 +34
> 1797595814951 +26
> 1797595814941 +10
> 1797595814939 +2
> 1797595814921 +18
> 1797595814897 +24
> 1797595814873 +24
> 1797595814863 +10
>
> Phil
>
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>  Phil Carmody wrote:
> 6 primes symmetrically arranged around a central prime
I didn't like the fast growth of the minimal solution so
> 683747 683759 683777 683783 683789 683807 683819
> . +12 +18 +6 +6 +18 +12
>
> 8 primes symmetrically arranged around a central prime
> 98303867 98303873 98303897 98303903 98303927 98303951 98303957 98303981
> 98303987
> . +6 +24 +6 +24 +24 +6 +24 +6
>
> 10 primes symmetrically arranged around a central prime
> 60335249851 +6
> 60335249857 +12
> 60335249869 +12
> 60335249881 +60
> 60335249941 +18
> 60335249959 +18
> 60335249977 +60
> 60335250037 +12
> 60335250049 +12
> 60335250061 +6
> 60335250067
>
> 12  well, that's a job for Jens :)
I searched a nonminimal instead:
Find 7 simultaneous primes in a specific chosen pattern
with 6 symmetric around the center.
Then see how far the symmetri extends.
After 338 cases with 10 symmetric, a 12 finally appeared:
3391781771953843 +/ 6, 24, 36, 66, 120, 126.
In Phil's notation:
3391781771953717 +6
3391781771953723 +54
3391781771953777 +30
3391781771953807 +12
3391781771953819 +18
3391781771953837 +6
3391781771953843 +6
3391781771953849 +18
3391781771953867 +12
3391781771953879 +30
3391781771953909 +54
3391781771953963 +6
3391781771953969
Prp testing by the GMP library and primality proving by PARI/GP.

Jens Kruse Andersen
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