Small but nontrivial prime puzzle

[MattcAnderson]

Hi math people,
The densest set of 'A' prime numbers in the interval [x,x+A] with x>A can be found with a prime constellation of length 'A'.
http://mathworld.wolfram.com/PrimeConstellation.html
These constellations have been tabulated and used for the Polymath8 project. See the tables at
http://www.opertech.com/primes/webdata/

An interval of 100 integers containing 24 primes can be found with the constellation
[0 4 6 12 16 24 30 34 40 42 46 52 54 60 66 70 72 76 82 84 90 94 96 100]
However, I do not know the primes.

Hope this is helpful.
Regards,
Matt

[fivemack]

[1187361391, 10980]
[1395350545, 10900]
[1395350595, 10844]
[1395350821, 10824]
[4143800871, 10598]
[7538609667, 10572]
[8668648953, 10542]

[fivemack]

Originally Posted by MattcAnderson

Hi math people,
The densest set of 'A' prime numbers in the interval [x,x+A] with x>A can be found with a prime constellation of length 'A'.
http://mathworld.wolfram.com/PrimeConstellation.html
These constellations have been tabulated and used for the Polymath8 project. See the tables at
http://www.opertech.com/primes/webdata/

Thanks for that pointer; I had found a summary table but not the actual offsets within the constellation. I can't convince myself that finding a /24 is remotely practical - I think it's at least millions of times harder than a /18, and finding a /18 took a year or so on 20% of 432 dual-P3 nodes at Zurich (http://www.asgard.ethz.ch/day/ws0102/waldvogel.phtml)

[fivemack]

On the principle that using lots of my own CPU time to cost myself more money is not as wise as wisdom itself, I've stopped at 11e9 (and updated the table upthread)

s=2^256
start=11*10^9
dmin=25000
q=vector(101,i,0)
p=nextprime(s+start);for(t=1,101,q[t]=p-s;p=nextprime(1+p))
r=101;xx=q[101];
q
while(true,\
  r2=1+r;if(r2==102,r2=1);\
   delta=q[r]-q[r2];\
   if(delta<dmin,print([q[r], delta]);dmin=delta);\
   xx=nextprime(xx+s+1)-s;\
   q[r2]=xx;\
   r=r2)

[LaurV]

I am still working on this (only in the evenings when I get home), with a different approach. Just to keep you informed.
Maybe expect some results in a week or so...

[fivemack]

Originally Posted by mart_r

Looking at the prime gap listings, the gap at 5378 is currently the best one below 2^256, so the next milestone would be a cluster with k=5376...

There are 64 primes in a 5376-range starting at 2^256+1187351356

[mart_r]

Originally Posted by fivemack

There are 64 primes in a 5376-range starting at 2^256+1187351356

Alas I don't think it's possible to squeeze out more than 73 primes in that range...

Anyway, to keep up with the competition, I'll offer k=10164 at s=2^256+1341068619664634010087.

[flagrantflowers]

Originally Posted by mart_r

Anyway, to keep up with the competition, I'll offer k=10164 at s=2^256+1341068619664634010087.

That's a big improvement.

[rajula]

I also wrote a program last evening and let it run for the night. This was found after 4h running time on one core (I will test more to see if I got lucky or if my approach was just better)

k = 9776

s = 115792089240225030208471517581734313024179825562764966188454037028443482485321 = 2^256 + (67 digits)

[bsquared]

I jumped on the bandwagon too. After an hour on one machine I have:

73603164627, 9624

I'll let it go overnight and see what's there in the morning.

[bsquared]

Originally Posted by bsquared

I jumped on the bandwagon too. After an hour on one machine I have:

73603164627, 9624

I'll let it go overnight and see what's there in the morning.

Best after 7 more hours is:

446184532951, 9514

... and proceeding at ~ 1e9 w/min. I'll let it go up to 1e12; in the meantime I'm playing a bit with a different approach.