Octoproths

[smh]

I tested n=141 with the old executable upto k=5,555*10^13

This is the closest i got:

43354856050725*2^141+1  is 3-PRP! (0.0004s+0.0004s)
43354856050725*2^141-1  is 3-PRP! (0.0002s+0.0038s)
43354856050725*2^(141+1)+1  is 3-PRP! (0.0002s+0.0022s)
43354856050725*2^(141+1)-1  is 3-PRP! (0.0002s+0.0021s)
2^141+43354856050725  is 3-PRP! (0.0001s+0.0021s)
2^141-43354856050725  is 3-PRP! (0.0001s+0.0030s)
2^(141+1)+43354856050725  is 3-PRP! (0.0001s+0.0023s)
2^(141+1)-43354856050725 is composite: [2FAAB440A92EB3DC] (0.0001s+0.0022s)

I'll try 135 next

[axn]

Originally Posted by smh

I'll try 135 next

If you are searching higher n's, try to search n = 4,7, or 10 (mod 15), esp n = 7 (mod 15). These are heavy weight n's. I think n=142 will make a good candidate.

[Templus]

Originally Posted by axn1

They differ only in the depth of the sieve, ie, the number of p's used to sieve. p < 10^5, 10^6 and 10^7 (resp. for fast, med & deep).

PS:- The numbers you posted are not octos. They are only prime for 2^n+/-k and 2^(n+1)+/-k. They are not prime for the other forms, k*2^n+/-1 and k*2^(n+1)+/-1

I am terribly sorry, I forgot to extend the ABC-line
I will test k=235 further on!

[smh]

Originally Posted by axn1

If you are searching higher n's, try to search n = 4,7, or 10 (mod 15), esp n = 7 (mod 15). These are heavy weight n's. I think n=142 will make a good candidate.

Okay, i'll test 142 then. I'm running the 'sieve' for a day or so on a p3 700 and see what pops up.

For 135 i didn't find any upto 1.196*10^13

12 had primes for the first 6 forms, of which 2 had also primes for the 7th form

[Dougy]

So far every test I have put the new program through was passed. So I believe it is working fine.

I ran the complete test on bases upto n=37. (and still going). And that base produced a whopping 83 octoproth primes.

Secondly, both bases 32 and 33 have no octoproth primes.

[Dougy]

When I put these (and others) into the batch factorisation:
2^39-540206575755
2^39-539552526135

and test for primality it says (even if i just type in the decimal too!)
9549238133 is composite
10203287753 is composite

However if I factorize them instead
9549238133 = 9549238133
10203287753 = 10203287753

implying they're prime.

This means that I will have missed some octoproths... But fortunately I kept the sieved files.

URL: http://www.alpertron.com.ar/ECM.HTM

[axn]

Originally Posted by Dougy

When I put these (and others) into the batch factorisation:
2^39-540206575755
2^39-539552526135

and test for primality it says (even if i just type in the decimal too!)
9549238133 is composite
10203287753 is composite

However if I factorize them instead
9549238133 = 9549238133
10203287753 = 10203287753

implying they're prime.

This means that I will have missed some octoproths... But fortunately I kept the sieved files.

URL: http://www.alpertron.com.ar/ECM.HTM

I too ran into this problem yesterday, while working with one of the lower n's (32, I think).

The "deep" version sieves upto p < 10^7, which means that for n <= 45 all the candidates will be automatically prime for 2^n+/-k forms!

You definitely need to recheck base 32 and 33!

[Dougy]

After rechecking the small bases, I've updated the text file again. It should be fixed. I've completed upto n=41. Some interesting properties...

Number of octoproth-primes for n=27,28,29,...
1,2,1,1,2,0,0,7,17,11,90,28,83,331,109,...

n=40 alone has 331 octoproth-primes To think only the other day I was hoping to break the 100 mark.

Number of k-values unsieved (octo_deep) for n=27,28,29,... over all possible k-values.
2,4,4,2,19,22,13,110,137,85,802,360,844,4434,1651,7552,...

[axn]

Originally Posted by Dougy

After rechecking the small bases, I've updated the text file again. It should be fixed. I've completed upto n=41. Some interesting properties...

Number of octoproth-primes for n=27,28,29,...
1,2,1,1,2,0,0,7,17,11,90,28,83,331,109,...

n=40 alone has 331 octoproth-primes To think only the other day I was hoping to break the 100 mark.

Number of k-values unsieved (octo_deep) for n=27,28,29,... over all possible k-values.
2,4,4,2,19,22,13,110,137,85,802,360,844,4434,1651,7552,...

Some missing values for n=32,33,34,35, and 37.

n = 32
---------
 409668105
 664495755
2368386195
2709707805
3383804865
3692088225
3762658725

n = 33
---------
 715414875
6876947175

n = 34
---------
  293705775
 1183281975
 1397861655
 3767954715
 4597935705
 8596001505

n=35
---------
17182250085
17783238795
20646922695
21811399155
22622064465
23416146075
24115395465
24449183535
25380028905

n=37
-----------
  7218568995
126139443165

n = 36,38,39, and 40 are fine.

Also, I have checked all high k's for n =27 thru 64 (high k's are k's that can't be reliably sieved because 2^n-k becomes smaller than the largest sieve prime). There are no hidden octo's in there

[Dougy]

Wow thanks, I didn't think there could be so many missing... it's all updated now.

So there might be octo's for all bases after 27. I've added the smallest octo for each base upto 71.

Most Wanted: n=72... searched k<1000000000000.

[smh]

I've found the following for n=142, searching K from 1 to 3.09x10^13

8444737373415*2^142+1  is 3-PRP! (0.0002s+0.0002s)
8444737373415*2^142-1  is 3-PRP! (0.0002s+0.0023s)
8444737373415*2^(142+1)+1  is 3-PRP! (0.0002s+0.0023s)
8444737373415*2^(142+1)-1  is 3-PRP! (0.0018s+0.0023s)
2^142+8444737373415  is 3-PRP! (0.0001s+0.0024s)
2^142-8444737373415  is 3-PRP! (0.0001s+0.0023s)
2^(142+1)+8444737373415  is 3-PRP! (0.0001s+0.0023s)
2^(142+1)-8444737373415 is 3-PRP! (0.0001s+0.0024s)

9532236817845*2^142+1  is 3-PRP! (0.0002s+0.0029s)
9532236817845*2^142-1  is 3-PRP! (0.0002s+0.0024s)
9532236817845*2^(142+1)+1  is 3-PRP! (0.0002s+0.0024s)
9532236817845*2^(142+1)-1  is 3-PRP! (0.0002s+0.0025s)
2^142+9532236817845  is 3-PRP! (0.0001s+0.0023s)
2^142-9532236817845  is 3-PRP! (0.0001s+0.0023s)
2^(142+1)+9532236817845  is 3-PRP! (0.0001s+0.0023s)
2^(142+1)-9532236817845 is 3-PRP! (0.0001s+0.0026s)

22732824274545*2^142+1  is 3-PRP! (0.0002s+0.0003s)
22732824274545*2^142-1  is 3-PRP! (0.0002s+0.0024s)
22732824274545*2^(142+1)+1  is 3-PRP! (0.0002s+0.0024s)
22732824274545*2^(142+1)-1  is 3-PRP! (0.0002s+0.0024s)
2^142+22732824274545  is 3-PRP! (0.0001s+0.0023s)
2^142-22732824274545  is 3-PRP! (0.0001s+0.0023s)
2^(142+1)+22732824274545  is 3-PRP! (0.0001s+0.0024s)
2^(142+1)-22732824274545 is 3-PRP! (0.0001s+0.0024s)

Close to 2 million numbers survived the sieve. Newpgen didn't make sence after this, since it removed candidates much slower than i was able to prp them.

I'll try 157 (another 7 mod 15) next.