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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}
Message 1 of 26 , Mar 4, 2003
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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 odd-yielding 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
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com
• ... Yeah, but even if the sums are expected 92.5% composite, and the products expected 96.25% composite, then randomly you d expect a single coincidence with
Message 2 of 26 , Mar 4, 2003
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--- Jon Perry <perry@...> wrote:
> '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 odd-yielding 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.

Yeah, but even if the sums are expected 92.5% composite, and the products
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 23-element
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

=====
"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, 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
Message 3 of 26 , Mar 4, 2003
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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
• ... run length 21 - (15776479..15776499) 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
Message 4 of 26 , Mar 4, 2003
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--- In primenumbers@yahoogroups.com, Jack Brennen <jack@b...> wrote:
>
> 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)

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.
• ... In which case I will stop mine, which is going at 8k/sec on this PC. But it agrees with your results so far.
Message 5 of 26 , Mar 4, 2003
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> run length 21 -> (15776479..15776499)
>
> 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.

In which case I will stop mine, which is going at 8k/sec on this PC. But it
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)=
Message 6 of 26 , Mar 4, 2003
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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+n-1);
vl=length(v);fl=0;
for (i=1,vl-1,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
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com
• Using: { test(n,k)= v=vector(k,i,i+n-1); vl=length(v);fl=0;pc=0; for (i=1,vl-1,for (j=i+1,vl, if (isprime(v[i]+v[j])!=isprime(v[i]*v[j]+1),fl=1;break); if
Message 7 of 26 , Mar 4, 2003
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Using:

{
test(n,k)=
v=vector(k,i,i+n-1);
vl=length(v);fl=0;pc=0;
for (i=1,vl-1,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
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com
• The first with rl 8 is [5255:2] [9312:1] is the only other to 10000. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/
Message 8 of 26 , Mar 4, 2003
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The first with rl 8 is

[5255:2]

[9312:1] is the only other to 10000.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-
• ... It correctly gave me the length 5 and 6 results, so I applied the strong law of small numbers, which I double-checked using engineers induction. That s
Message 9 of 26 , Mar 4, 2003
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--- Jack Brennen <jack@...> wrote:
> 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.

It correctly gave me the length 5 and 6 results, so I applied the strong law
of small numbers, which I double-checked 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

=====
"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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• ... 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 -
Message 10 of 26 , Mar 4, 2003
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--- In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:
>
> 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.
• 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,
Message 11 of 26 , Mar 4, 2003
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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
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com
• ... 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
Message 12 of 26 , Mar 4, 2003
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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

=====
"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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• ... I agree with you on all previous data points, and this one too:
Message 13 of 26 , Mar 4, 2003
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--- "jbrennen <jack@...>" <jack@...> wrote:
> --- In primenumbers@yahoogroups.com, Jack Brennen <jack@b...> wrote:
> >
> > 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)

I agree with you on all previous data points, and this one too:
<<<
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
>>>

> 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.

The "t=60.45" means the program took 60.45 seconds of 900MHz Duron.
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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• ... unbleh!
Message 14 of 26 , Mar 4, 2003
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--- Phil Carmody <thefatphil@...> wrote:
> <<<
> 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.

unbleh!

<<<
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

=====
"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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• ... 370902159..370902183 is len 25 (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
Message 15 of 26 , Mar 4, 2003
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> 38392415..38392436 is len 22
> 76350646..76350668 is len 23
> 76350646..76350669 is len 24

370902159..370902183 is len 25

(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

=====
"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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• ... 3330 further seconds without the prime requirement provided: 1091876084..1091876109 is len 26 ... Agreed. 5901351..5901367 is len 17 (16s) 6120922..6120939
Message 16 of 26 , Mar 5, 2003
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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:
> >
> > 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.

Agreed.

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 re-run
them all, and push it further. 250000/s just isn't fast enough... ;-)

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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• 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
Message 17 of 26 , Mar 5, 2003
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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
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com
• ... You seem back-to-front, Jon. 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
Message 18 of 26 , Mar 5, 2003
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--- Jon Perry <perry@...> wrote:
> '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.

You seem back-to-front, Jon.

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
> at...

Nope, trees and forests with loops are frowned upon (and any other DAGs, of
course)

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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• ... 311606294..311606316 is len 23 (instant, doh!) 467282388..467282411 is len 24 (9m50s) 1142221381..1142221405 is len 25 (42m33s) It was only 4% faster
Message 19 of 26 , Mar 5, 2003
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--- Phil Carmody <thefatphil@...> wrote:
> > 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)
> 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)

311606294..311606316 is len 23 (instant, doh!)
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

=====
"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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• These are for the isomorph a+b ab-1. (no guarenteed prime) run length 1 [1,...,1] run length 2 [2,...,3] run length 3 [5,...,7] run length 4 [56,...,59]
Message 20 of 26 , Mar 5, 2003
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These are for the isomorph a+b <=> ab-1.

(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
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com
• ... First ones with primes appended, and appended. ... 787374..787387 is len 14 1262441..1262455 is len 15 3623705..3623720 is len 16 7993838..7993854 is len
Message 21 of 26 , Mar 5, 2003
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--- Jon Perry <perry@...> wrote:
> These are for the isomorph a+b <=> ab-1.
>
> (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)

=====
"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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• ... 535716000..535716021 is len 22 (27m35s) 535716000..535716022 is len 23 I stopped it at ~1.4G after nearly an hour looking for a 24. Phil
Message 22 of 26 , Mar 5, 2003
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--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@y...> wrote:
> --- Jon Perry <perry@g...> wrote:
> > These are for the isomorph a+b <=> ab-1.
> >
> > (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..535716021 is len 22 (27m35s)
535716000..535716022 is len 23

I stopped it at ~1.4G after nearly an hour looking for a 24.

Phil
• 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
Message 23 of 26 , Mar 31, 2003
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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
all-composite sets seemed to be too easy; although we did not manage a proof
of the existence of arbitary sets as attempted by polytope.

The sets mentions are not the first composite-only runs:

And