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Re: [PrimeNumbers] Isomorphic Numbers
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 Jon Perry <perry@...> wrote:> Prompted by a thread Jack and myself had a while ago, you will recall that
Notcontinuous, I present
> the set:
>
> S={1,2,3,4,5}
>
> has the property that a+b is prime <=> a.b+1 is prime
>
> Can anyone find a larger set of continuous (or not) integers with this
> property?
{ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,
41, 43, 45, 47, 49, ....
You get the idea, I trust.
Phil
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 Jon Perry <perry@...> wrote:> Prompted by a thread Jack and myself had a while ago, you will recall that
{4,5,6,7,8,9} seems to do the business for contiguous ranges.
> the set:
>
> S={1,2,3,4,5}
>
> has the property that a+b is prime <=> a.b+1 is prime
>
> Can anyone find a larger set of continuous (or not) integers with this
> property?
Spoilt, as ever, by the Carmodyprime 91.
These I didn't use just pen and paper for, and therefore are much more
likely to be wrong 
7 : {684,685,686,687,688,689,690}
8 : {1081,1082,1083,1084,1085,1086,1087,1088}
9 : {1872,...1880}
11: {2419,...2429}
12: {9811,...9822}
13: {30650,...30662}
14: {64027,...64040}
15: {64029,...64043}
16: {102035,...102050}
17: {146467,...146483}
18: {372879,...372896}
22: {613787,...613808}
22 elements is 11*11 oddyielding pairs. So there's 121 coincidences in the
above set. That can't be right, surely?
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
"Are you still bombing your wife?"  Winjer
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'7 : {684,685,686,687,688,689,690}
8 : {1081,1082,1083,1084,1085,1086,1087,1088}
9 : {1872,...1880}
11: {2419,...2429}
12: {9811,...9822}
13: {30650,...30662}
14: {64027,...64040}
15: {64029,...64043}
16: {102035,...102050}
17: {146467,...146483}
18: {372879,...372896}
22: {613787,...613808}
22 elements is 11*11 oddyielding pairs. So there's 121 coincidences in the
above set. That can't be right, surely? '
I suppose the 'coincidence level' depends on the proportion of primes in
each range.
Jon Perry
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 Jon Perry <perry@...> wrote:> '7 : {684,685,686,687,688,689,690}
Yeah, but even if the sums are expected 92.5% composite, and the products
> 8 : {1081,1082,1083,1084,1085,1086,1087,1088}
> 9 : {1872,...1880}
> 11: {2419,...2429}
> 12: {9811,...9822}
> 13: {30650,...30662}
> 14: {64027,...64040}
> 15: {64029,...64043}
> 16: {102035,...102050}
> 17: {146467,...146483}
> 18: {372879,...372896}
> 22: {613787,...613808}
>
> 22 elements is 11*11 oddyielding pairs. So there's 121 coincidences in the
> above set. That can't be right, surely? '
>
> I suppose the 'coincidence level' depends on the proportion of primes in
> each range.
expected 96.25% composite, then randomly you'd expect a single coincidence
with probability 0.893 and therefore 121 of them with probability 1/870200.
OK, it looks more possible than I first thought, but still, the 23element
set seems too elusive (requiring 132 coincidences). I ran it for 20 times
longer than the 22 above. I can only assume that I searched 20* further, but
I didn't log how far I got. I was using GP/Pari, so a C/Pascal
implementation should race through the possibilities much faster.
It's surely someone else's turn to find one. Jack? Jens? Jud? Jon?
Other people beginning with a J?
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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Phil, I think there is a problem with your sets of numbers... you don't
seem to have considered the case where a & b are both even, thus
requiring that ab+1 be an odd composite.
For instance, you gave the range (684..690), but we have:
684+688 is composite
684*688+1 is prime
But some of the ranges you gave are correct, by coincidence I guess. :)
I get the following results up to length 20, which took some 2 minutes
of CPU time using C with the GMP library... Note that someone might
want to doublecheck these, since my primality testing wasn't rigorous,
only probable:
run length 2 > (1..2)
run length 3 > (1..3)
run length 4 > (1..4)
run length 5 > (1..5)
run length 6 > (4..9)
run length 7 > (906..912)
run length 8 > (1081..1088)
run length 9 > (2212..2220)
run length 10 > (2419..2428)
run length 11 > (2419..2429)
run length 12 > (20153..20164)
run length 13 > (30650..30662)
run length 14 > (64027..64040)
run length 15 > (64029..64043)
run length 16 > (102035..102050)
run length 17 > (372879..372895)
run length 18 > (1060846..1060863)
run length 19 > (4895741..4895759)
run length 20 > (5385547..5385566)
There is no run of length 21 below 10^7 (again, subject to the
caveat of only using a probable prime test).
Jack 0 Attachment
 In primenumbers@yahoogroups.com, Jack Brennen <jack@b...> wrote:>
run length 21 > (15776479..15776499)
> There is no run of length 21 below 10^7 (again, subject to the
> caveat of only using a probable prime test).
>
No run of length 22 below 2.8*10^7.
I'll let it run for a day or so before stopping the search.
It's currently extending the search limit by about 14000/second. 0 Attachment
> run length 21 > (15776479..15776499)
In which case I will stop mine, which is going at 8k/sec on this PC. But it
>
> No run of length 22 below 2.8*10^7.
>
> I'll let it run for a day or so before stopping the search.
> It's currently extending the search limit by about 14000/second.
agrees with your results so far.
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'Note that someone might
want to doublecheck these, since my primality testing wasn't rigorous,
only probable:'
They are all valid:
{
test(n,k)=
v=vector(k,i,i+n1);
vl=length(v);fl=0;
for (i=1,vl1,for (j=i+1,vl,
if (isprime(v[i]+v[j])!=isprime(v[i]*v[j]+1),fl=1;break);
if (fl==1,break)));
if (fl==0,print("ok"),print("fail"));
}
test(906,7)
test(1081,8)
test(2212,9)
test(2419,10)
test(2419,11)
test(20153,12)
test(30650,13)
test(64027,14)
test(64029,15)
test(102035,16)
test(372879,17)
test(1060846,18)
test(4895741,19)
test(5385547,20)
test(15776479,21)
Jon Perry
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Using:
{
test(n,k)=
v=vector(k,i,i+n1);
vl=length(v);fl=0;pc=0;
for (i=1,vl1,for (j=i+1,vl,
if (isprime(v[i]+v[j])!=isprime(v[i]*v[j]+1),fl=1;break);
if (fl==1,break);
if (isprime(v[i]+v[j]),pc++)));
[fl,pc];
}
to check the current list yields 0 primes for all run length>7.
So I think we should adopt a policy of a least 1 prime from a range.
Jon Perry
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The first with rl 8 is
[5255:2]
[9312:1] is the only other to 10000.
Jon Perry
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 Jack Brennen <jack@...> wrote:> Phil, I think there is a problem with your sets of numbers... you don't
It correctly gave me the length 5 and 6 results, so I applied the strong law
> seem to have considered the case where a & b are both even, thus
> requiring that ab+1 be an odd composite.
of small numbers, which I doublechecked using engineers' induction.
That's what caveats are for, I guess.
I shut down my emacs/shell/GP window, so I have no way of working out where
I went wrong, it was all ephemeral.
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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 In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:>
In that case:
> So I think we should adopt a policy of a least 1 prime from a range.
>
run length 8 > (5255..5262)
run length 9 > (8009..8017)
run length 10 > (16268..16277)
run length 11 > (16268..16278)
run length 12 > (118151..118162)
run length 13 > (252035..252047)
run length 14 > (267485..267498)
run length 15 > (267485..267499)
run length 16 > (305923..305938)
With no length 17 run less than 5*10^6. 0 Attachment
Another challenge is to find to max. number of primes.
rl8;
? for (n=10000,50000,x=test(n,8);if (!x[1] && x[2]>0,print1(n":"x[2]", ")))
12106:2, 12775:2, 13069:2, 15869:3, 15870:3, 16064:1, 16189:2, 16268:1,
17409:1,
17920:2, 17924:2, 18344:5, 18411:2, 18416:1, 18552:1, 19466:1, 21715:2,
21806:1
, 22820:1, 24689:1, 25956:1, 26113:2, 26865:1, 27304:3, 28479:3, 28480:2,
28481:
1, 29709:1, 29931:1, 31614:1, 31725:1, 34035:1, 34410:1, 34805:1, 35091:1,
35359
:2, 36204:1, 36205:2, 36633:2, 36634:3, 36716:1, 36784:2, 37520:1, 38829:1,
3924
7:2, 39296:1, 39759:1, 39760:2, 39935:2, 40403:3, 40404:3, 40748:1, 42866:1,
434
05:2, 43829:1, 45192:1, 46362:1, 47678:1, 48019:3, 48315:1, 49484:1,
49755:1, 49
789:2, 49790:1, 49953:1,
So 18344:5 is the current record.
Jon Perry
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 Jack Brennen <jack@...> wrote:> run length 10 > (2419..2428)
2420+2420 = 4840 = composite
2420*2420+1=5856401 = prime
Are the two elements supposed to be distinct? Jon doesn't state
that in his frist post today.
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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 "jbrennen <jack@...>" <jack@...> wrote:>  In primenumbers@yahoogroups.com, Jack Brennen <jack@b...> wrote:
I agree with you on all previous data points, and this one too:
> >
> > There is no run of length 21 below 10^7 (again, subject to the
> > caveat of only using a probable prime test).
> >
>
> run length 21 > (15776479..15776499)
<<<
Starting program: /mnt/net/kilospaz/home/phil/projects/maths/primes/conjectures/perry/sumprod 1 21
[...]
t=60.450000 b=15728652)
Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
(gdb) next
(gdb)
(gdb)
15776479..15776499 is len 21>>>
The "t=60.45" means the program took 60.45 seconds of 900MHz Duron.
> No run of length 22 below 2.8*10^7.
>
> I'll let it run for a day or so before stopping the search.
> It's currently extending the search limit by about 14000/second.
I'm currently clocking 200000/s, but expect to slow down as time goes on.
<<<
t=89.730000 b=37748755)
Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
(gdb) next
(gdb)
(gdb)
38392415..38392436 is len 22>>>
90s, but that run didn't start from 0, but from the len=21 one.
So I'm looking at the 23 now, hopefully I'll be able to finish my sentence
before it find one...
..., bleh.
Phil
=====
"Only an admission that he does possess weapons of mass destruction
would do, sources said: 'The rest is just gesture politics."  Hoon
"Are you still bombing your wife?"  Winjer
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 Phil Carmody <thefatphil@...> wrote:> <<<
unbleh!
> t=89.730000 b=37748755)
>
> Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
> (gdb) next
> (gdb)
> (gdb)
> 38392415..38392436 is len 22
> >>>
>
> 90s, but that run didn't start from 0, but from the len=21 one.
>
> So I'm looking at the 23 now, hopefully I'll be able to finish my sentence
> before it find one...
>
> ..., bleh.
<<<
t=156.960000 b=75497473)
Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
(gdb) next
(gdb)
(gdb)
76350646..76350668 is len 23>>>
and instantly thereafter:
76350646..76350669 is len 24
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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> 38392415..38392436 is len 22
370902159..370902183 is len 25
> 76350646..76350668 is len 23
> 76350646..76350669 is len 24
(1262 seconds, FWIW)
I'll not look at the 'must include a prime' version until I've got to 2^31
and my signed ints run out of juice.
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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 "jbrennen <jack@...>" <jack@...> wrote:
3330 further seconds without the prime requirement provided:
1091876084..1091876109 is len 26
>  In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:
Agreed.
> >
> > So I think we should adopt a policy of a least 1 prime from a range.
> >
>
> In that case:
>
> run length 8 > (5255..5262)
> run length 9 > (8009..8017)
> run length 10 > (16268..16277)
> run length 11 > (16268..16278)
> run length 12 > (118151..118162)
> run length 13 > (252035..252047)
> run length 14 > (267485..267498)
> run length 15 > (267485..267499)
> run length 16 > (305923..305938)
>
> With no length 17 run less than 5*10^6.
5901351..5901367 is len 17 (16s)
6120922..6120939 is len 18 (<1s)
15736249..15736267 is len 19 (35s)
15736249..15736268 is len 20 (instant)
75952831..75952851 is len 21 (3m49s)
311606294..311606315 is len 22 (15m9s)
I'm just going to break the code^W^W^Wdo a few tweaks, and then I'll rerun
them all, and push it further. 250000/s just isn't fast enough... ;)
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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'Are the two elements supposed to be distinct? Jon doesn't state
that in his first post today.'
This wasn't a condition in the original discussion, indeed we have 4+4=8,
4*4+1=17, same for 2.
For stringency this could be added, although graphs with loops are frowned
at...
Jon Perry
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 Jon Perry <perry@...> wrote:> 'Are the two elements supposed to be distinct? Jon doesn't state
You seem backtofront, Jon.
> that in his first post today.'
>
> This wasn't a condition in the original discussion, indeed we have 4+4=8,
> 4*4+1=17, same for 2.
8 is composite, 17 is prime and {1,2,3,4,5} was provided as a solution.
Therefore the condition that the pair's elements are distinct _was_ a
condition in the first one?
If it wasn't a condition, then {1,2,3,4,5} wouldn't have been a solution.
> For stringency this could be added, although graphs with loops are frowned
Nope, trees and forests with loops are frowned upon (and any other DAGs, of
> at...
course)
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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 Phil Carmody <thefatphil@...> wrote:> > run length 8 > (5255..5262)
311606294..311606316 is len 23 (instant, doh!)
> > run length 9 > (8009..8017)
> > run length 10 > (16268..16277)
> > run length 11 > (16268..16278)
> > run length 12 > (118151..118162)
> > run length 13 > (252035..252047)
> > run length 14 > (267485..267498)
> > run length 15 > (267485..267499)
> > run length 16 > (305923..305938)
> 5901351..5901367 is len 17 (16s)
> 6120922..6120939 is len 18 (<1s)
> 15736249..15736267 is len 19 (35s)
> 15736249..15736268 is len 20 (instant)
> 75952831..75952851 is len 21 (3m49s)
> 311606294..311606315 is len 22 (15m9s)
467282388..467282411 is len 24 (9m50s)
1142221381..1142221405 is len 25 (42m33s)
It was only 4% faster looking for ones with primes in the range. So I think
that for me it would make sense to look for all such runs, regardless of the
existance of primes in the sum's range.
Phil
=====
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would do, sources said: 'The rest is just gesture politics."  Hoon
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These are for the isomorph a+b <=> ab1.
(no guarenteed prime)
run length 1 [1,...,1]
run length 2 [2,...,3]
run length 3 [5,...,7]
run length 4 [56,...,59]
run length 5 [121,...,125]
run length 6 [211,...,216]
run length 7 [819,...,825]
run length 8 [1470,...,1477]
run length 9 [2231,...,2239]
run length 10 [15886,...,15895]
run length 11 [44275,...,44285]
run length 12 [44275,...,44286]
run length 13 [44275,...,44287]
I left the code running for about an hour after I found rl13, so rl14 is
high.
Jon Perry
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 Jon Perry <perry@...> wrote:> These are for the isomorph a+b <=> ab1.
First ones with primes appended, and appended.
>
> (no guarenteed prime)
> run length 1 [1,...,1]
787374..787387 is len 14
> run length 2 [2,...,3]
> run length 3 [5,...,7]
> run length 4 [56,...,59]
> run length 5 [121,...,125] 131..135
> run length 6 [211,...,216]
> run length 7 [819,...,825] 915..921
> run length 8 [1470,...,1477] 1470..1477
> run length 9 [2231,...,2239] 2231..2239
> run length 10 [15886,...,15895] 42672..42681
> run length 11 [44275,...,44285] 89663..89673
> run length 12 [44275,...,44286] 89663..89674
> run length 13 [44275,...,44287] 505989..506001
1262441..1262455 is len 15
3623705..3623720 is len 16
7993838..7993854 is len 17
7993838..7993855 is len 18
7993838..7993856 is len 19
7993838..7993857 is len 20
117532072..117532092 is len 21 (6m40s)
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 In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@y...> wrote:>  Jon Perry <perry@g...> wrote:
535716000..535716021 is len 22 (27m35s)
> > These are for the isomorph a+b <=> ab1.
> >
> > (no guarenteed prime)
>
> First ones with primes appended, and appended.
>
> > run length 1 [1,...,1]
> > run length 2 [2,...,3]
> > run length 3 [5,...,7]
> > run length 4 [56,...,59]
> > run length 5 [121,...,125] 131..135
> > run length 6 [211,...,216]
> > run length 7 [819,...,825] 915..921
> > run length 8 [1470,...,1477] 1470..1477
> > run length 9 [2231,...,2239] 2231..2239
> > run length 10 [15886,...,15895] 42672..42681
> > run length 11 [44275,...,44285] 89663..89673
> > run length 12 [44275,...,44286] 89663..89674
> > run length 13 [44275,...,44287] 505989..506001
>
> 787374..787387 is len 14
> 1262441..1262455 is len 15
> 3623705..3623720 is len 16
> 7993838..7993854 is len 17
> 7993838..7993855 is len 18
> 7993838..7993856 is len 19
> 7993838..7993857 is len 20
> 117532072..117532092 is len 21 (6m40s)
535716000..535716022 is len 23
I stopped it at ~1.4G after nearly an hour looking for a 24.
Phil 0 Attachment
Looking at:
'Check out his idea here:
http://mnemo.nu/math/problems/?action=problem_view&id=107'
I can't see how polytope has proved that all ab+1 are composites.
Indeed using this code:
{
forstep (n=4,20,2,v=vector(n/2);
for (i=1,n/2,v[i]=n!/2+i);
for (i=1,n/2,for (j=i+1,n/2,
if (isprime(v[i]*v[j]+1),print(n":"i","j)))))
}
yields the following exceptions.
16:4,7
18:3,6
18:5,8
20:7,10
20:8,9
(note 16:4,7 implies that (16!/2+4) * (16!/2+7) + 1 =
109440784174463838480384029 is prime)
We did adopt a policy of at least one prime in the thread, as the
allcomposite sets seemed to be too easy; although we did not manage a proof
of the existence of arbitary sets as attempted by polytope.
http://groups.yahoo.com/group/primenumbers/message/11660
The sets mentions are not the first compositeonly runs:
http://groups.yahoo.com/group/primenumbers/message/11670
And
http://groups.yahoo.com/group/primenumbers/message/11673
sums up to current knowledge of the problem.
Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
http://www.users.globalnet.co.uk/~perry/DIVMenu/
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