## Interesting prime sequence

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• As an interesting little project, I ve been looking for sequences of consecutive integers where N+1 is prime, N+2 is 2 times a prime, N+3 is 3 times a prime,
Message 1 of 6 , Mar 27, 2002
As an interesting little project, I've been looking for
sequences of consecutive integers where N+1 is prime,
N+2 is 2 times a prime, N+3 is 3 times a prime, etc.

Another way to look at it is to find an integer N that
maximizes the number of primes in the sequence
N+1, N/2+1, N/3+1, N/4+1, ...

I've been searching for the smallest such N for each
successive length:

Length N
------ -
1 1
2 2
3 12
4 12720
5 19440
6 5516280
7 5516280
8 7321991040
9 363500177040
10 2394196081200
11 3163427380990800

I'm currently sieving through about 7 million candidates/second
while looking for the smallest such sequence of length 12.
I'm sure that sieve maestro Phil could beat that quite easily,
with his Alpha sieve programming skills, but it sure beats
doing trial division :-)
• ... should obviously be highlighted, and postulated upon such that an infinite number of such pairs exist. Jon Perry perry@globalnet.co.uk
Message 2 of 6 , Mar 27, 2002
> 6 5516280
> 7 5516280

should obviously be highlighted, and postulated upon such that an infinite
number of such pairs exist.

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

-----Original Message-----
From: jbrennen [mailto:jack@...]
Sent: 27 March 2002 20:37

As an interesting little project, I've been looking for
sequences of consecutive integers where N+1 is prime,
N+2 is 2 times a prime, N+3 is 3 times a prime, etc.

Another way to look at it is to find an integer N that
maximizes the number of primes in the sequence
N+1, N/2+1, N/3+1, N/4+1, ...

I've been searching for the smallest such N for each
successive length:

Length N
------ -
1 1
2 2
3 12
4 12720
5 19440
6 5516280
7 5516280
8 7321991040
9 363500177040
10 2394196081200
11 3163427380990800

I'm currently sieving through about 7 million candidates/second
while looking for the smallest such sequence of length 12.
I'm sure that sieve maestro Phil could beat that quite easily,
with his Alpha sieve programming skills, but it sure beats
doing trial division :-)

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• ... You need to send this to Sloane. +---------------------------------------------------------+ ...
Message 3 of 6 , Mar 27, 2002
At 08:36 PM 3/27/2002 +0000, jbrennen wrote:

>I've been searching for the smallest such N for each
>successive length:

You need to send this to Sloane.

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+
• ... So basically one sieves out -1...-L (mod p) for each p not a factor of any 1..L. And one sieves _in_ 0 mod p otherwise? Sounds, erm, doable... Phil
Message 4 of 6 , Mar 27, 2002
--- jbrennen <jack@...> wrote:
> As an interesting little project, I've been looking for
> sequences of consecutive integers where N+1 is prime,
> N+2 is 2 times a prime, N+3 is 3 times a prime, etc.
>
> Another way to look at it is to find an integer N that
> maximizes the number of primes in the sequence
> N+1, N/2+1, N/3+1, N/4+1, ...
>
> I've been searching for the smallest such N for each
> successive length:
>
> Length N
> ------ -
> 1 1
> 2 2
> 3 12
> 4 12720
> 5 19440
> 6 5516280
> 7 5516280
> 8 7321991040
> 9 363500177040
> 10 2394196081200
> 11 3163427380990800
>
>
> I'm currently sieving through about 7 million candidates/second
> while looking for the smallest such sequence of length 12.
> I'm sure that sieve maestro Phil could beat that quite easily,
> with his Alpha sieve programming skills, but it sure beats
> doing trial division :-)

So basically one sieves out -1...-L (mod p) for each p not a factor
of any 1..L. And one sieves _in_ 0 mod p otherwise?

Sounds, erm, doable...

Phil

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• ... Alright, just for you I ran on an x86 machine...
Message 5 of 6 , Mar 28, 2002
--- jbrennen <jack@...> wrote:
> As an interesting little project, I've been looking for
> sequences of consecutive integers where N+1 is prime,
> N+2 is 2 times a prime, N+3 is 3 times a prime, etc.
>
> Another way to look at it is to find an integer N that
> maximizes the number of primes in the sequence
> N+1, N/2+1, N/3+1, N/4+1, ...
>
> I've been searching for the smallest such N for each
> successive length:
>
> Length N
> ------ -
> 1 1
> 2 2
> 3 12
> 4 12720
> 5 19440
> 6 5516280
> 7 5516280
> 8 7321991040
> 9 363500177040
> 10 2394196081200
> 11 3163427380990800
>
>
>
> I'm currently sieving through about 7 million candidates/second
> while looking for the smallest such sequence of length 12.
> I'm sure that sieve maestro Phil could beat that quite easily,
> with his Alpha sieve programming skills, but it sure beats
> doing trial division :-)

Alright, just for you I ran on an x86 machine...

<<<
phil@nonospaz:brennen\$ time ../gensv.nono brennen12.gf -s10000 -w300
| ./brennent 12
Verifying brennen 12
#I ZIP: 6 primes, relation = 0 mod 30030 (expanded)
#I SPESH: 4 primes, reduction = 215441 -> 6545
#I SMALL@2: 4 dumb, 6 quick to 71
#I PAUL: 190 primes <=1291
#I WHEEL: 300 other primes <= 3643
#I FACTOR: 2.442830e-11
#I Flags: INFO MARKS
BRENNEN 12 22755817971366480

^C
real 24m51.336s
user 11m51.350s
sys 0m0.560s
>>>

All with 1 Fermat, so a few quick more MRs (this is a small number, I
think that 10 MRs is enough!

<<<
bash-2.05a\$ calc
C-style arbitrary precision calculator (version 2.11.5t4.5)
[Type "exit" to exit, or "help" for help.]

> p=22755817971366480
> for(i=1; i<=12; ++i) {
>> print ptest((p+i)/i)
>> }
1
1
1
1
1
1
1
1
1
1
1
1
>>>

However, it's not all peachy, the 20 minutes I spent coding the
tester has left some bugs, as it claimed another result that only had
11 real terms...
ARGH - out by one error (loops from 1..12 are passed when the counter
reaches 13!!!!)

I did overrun quite a way before I noticed the result - shall we call
it 8 minutes CPU time?

Hmmm -
10: 2394196081200
11: 3163427380990800
12: 22755817971366480
Looks like either the 11 result is too high (from a moral obligation
sense), or my 12 result is a pretender. To be honest, I didn't even
run a quick test on <12 to see if the code works, so I've not
verified that I find the same results as yours.
However, I'm using the command-line/pipe version of gensv which has
been stable for several months now, so it's only my test code that
can have bugs (I hope).

13 is of course going to be substantially harder.

Phil

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• Have you thought about extending this idea: 1)N+k is prime, N+2k is 2*{a prime}, etc... 2)N+1 is prime, N+2 is k*{a prime}, N+3 is 2*k*{a prime} 3)N+j is
Message 6 of 6 , Apr 3, 2002
Have you thought about extending this idea:

1)N+k is prime, N+2k is 2*{a prime}, etc...
2)N+1 is prime, N+2 is k*{a prime}, N+3 is 2*k*{a prime}
3)N+j is prime, N+2j is k*{a prime}, N+3j is 2*k*{a prime}

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

-----Original Message-----
From: jbrennen [mailto:jack@...]
Sent: 27 March 2002 20:37

As an interesting little project, I've been looking for
sequences of consecutive integers where N+1 is prime,
N+2 is 2 times a prime, N+3 is 3 times a prime, etc.

Another way to look at it is to find an integer N that
maximizes the number of primes in the sequence
N+1, N/2+1, N/3+1, N/4+1, ...

I've been searching for the smallest such N for each
successive length:

Length N
------ -
1 1
2 2
3 12
4 12720
5 19440
6 5516280
7 5516280
8 7321991040
9 363500177040
10 2394196081200
11 3163427380990800

I'm currently sieving through about 7 million candidates/second
while looking for the smallest such sequence of length 12.
I'm sure that sieve maestro Phil could beat that quite easily,
with his Alpha sieve programming skills, but it sure beats
doing trial division :-)

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