Search for sexy or octy prime

[Joshua2]

5897131077*2^34000-1 is prime! Time : 1.535 sec.
5897131077*2^34000+1 is prime! Time: 1.545 sec.
has over 10,000 digits.

Im doing a search on n=3400. I sieved for 4-tuplet with newpgen up to 10B. None. I found a twin though. So I now have a list of -1 numbers that are prime. I want to do prp on the +3 and +5 ones. Then I will want test all the +1 numbers and those ones that are prime, I will test the +3 and +5 for those. I have a txt file that has just the lines with "is prime!" or a plus ("+") sign.

[Cybertronic]

+3, I understand correct ? If n -1 / +1 are primes , so n+3 nerver a prime.
Just -7,-5,+5,+7. n+3 is divisible by 3.

[Joshua2]

Yes, I figured out I am doing +5 and +7 and -5 and -7 actually. I did 10G on n=34000 and now am doing 110G on n=34030. I actually got the world's largest cousin prime, and now am looking for the world's largest sexy prime. Does anyone know what the world's largest octy prime is, or even what it is called?

[Jens K Andersen]

I don't think you got the world's largest cousin prime. The proven record has 10154 digits but your form is not easily provable. The prp record has 11311 digits which is more than your form.
I have not heard of anybody searching for or naming primes 8 apart.

[Joshua2]

Well, I proved the LLR form, and I am primo'ing the other with 3.0.6 modified to accept such large numbers (~10300 digitis). I emailed F. Morain as well. It shouldn't be too hard to prove with a distributed fastECPP. Does Morain have the only one, and has anyone asked if he will share it?

[Jens K Andersen]

I don't know a publicly available version of distributed ECPP, but using ECPP on a cousin prime sounds like a waste of cpu time. Even if you already have a proven prime and a prp, it would be easier to start over on an easily provable form with PFGW, like the current proven record by Alm, Fleuren and I. It would probably take me less than 2 weeks to find both a proven cousin prime and a proven sexy prime at your size on my 2.4 GHz Core 2 Duo.
I have only heard about one Primo attempt above 10000 digits and that failed.

[Joshua2]

Oh, cool. I didn't know that when I started I would have done it that way probably. I thought only k*2^n +-1 were easily provable or k*primorial(n) +-1. Would octy or 4-tuples be the largest thing easily provable then? Are primorial proofs much slower than llr?

[Jens K Andersen]

For any even n, there are easily provable forms for two primes n apart. But for small n>2, I have not heard of known forms with an optimized prp test like k*2^n+/-1. A generic prp test is required. PFGW can make a prp test of any number but a generic PFGW test takes several times longer than llr on an optimized form. For two primes in a fixed pattern like cousin and sexy primes, accepting the slower prp tests is better in the end than doing fast tests but then having to run ECPP. If the goal is three primes in a fixed pattern like a prime triplet, then fast prp testing with a program like llr can become more important than avoiding ECPP. There are easily provable forms for prime triplets, quadruplets and quintuplets. There are not for sextuplets but the sextuplet record is only 427 digits so needing ECPP certainly doesn't matter there. The PFGW proofs of these easily provable forms take much longer than an llr proof but much shorter than finding a single prp of that size, so the proving time doesn't matter in practice.

[Joshua2]

Ok, I found, the world's largest sexy prime. Francois Morain verified the +7, and LLR verified the +1. Is there a prime pages slot for it, or just update the wikipedia article?

[Jens K Andersen]

The Prime Pages does not register sexy primes, but Morain's ECPP proof will qualify for the Top-20 ECPP.

Wikipedia requires a published reliable source. The +1 prime is easy to verify. If the +7 prime becomes listed in the Prime Pages and you post an announcement here then I think it's good enough to mention the sexy prime in Wikipedia although forum postings are largely not acceptable as sources. But in order to state in Wikipedia that it is the largest known sexy prime, a source like http://mathworld.wolfram.com/SexyPrimes.html would be good. My former sexy record is currently listed there. I may run a search on an easily provable form to retake the record. As mentioned earlier in this thread, it is relatively easy to beat your record with no need for ECPP (I wonder whether Morain was told that).

ECPP would be needed for all the following prp's but I have no plans to run ECPP on any of them or ask others to do it. All prp's found by me were found with PrimeForm/GW.

As far as I can tell, the largest known sexy prp's are:
12^16392-13 and 12^16392-7, 17690 digits, found by Lelio R Paula in 2006.

I have tested p-6 and p+6 for a lot of larger known primes and prp's, but no luck.
Here are three smaller sexy prp's between the proven and prp record, together with the discoverer of the used prime or prp:
865906^2048+1+6, 12160 digits, Herranen, Proth.exe
13719*2^42371+1-6, 12760 digits, Ballinger, Proth.exe
2^53406+577294575-6, 16077 digits, Lelio R Paula

I had more luck with large cousin prp's:
207609*2^42023+1+4, 12656 digits, +1 prime by Meunier, Proth.exe
634031143*2^66406+1-4, 20000 digits, +1 by Kostolansky, Proth.exe
11199*2^66751+1+4, 20099 digits, +1 by Brazier, Proth.exe
474435381*2^98394-1-4, 29629 digits, -1 by Angel, Jobling, Augustin, NewPGen, OpenPFGW

In 2007 Ken Davis used NewPGen and PrimeForm/GW to find 28000+ 10043-digit primes of form k*2^33333+1 for an AP4 search mentioned in http://tech.groups.yahoo.com/group/p...m/message/8762. He gave me access to the primes, and I have used them to find a gigantic prime/prp gap of all even sizes from 8 to 100, except size 62. Here are the found prp's:

89223969*2^33333+1-8
180855726*2^33333+1+10
75065573*2^33333+1+12
71918921*2^33333+1+14
91186818*2^33333+1+16
89541420*2^33333+1+18
50066630*2^33333+1+20
95908248*2^33333+1+22
55428699*2^33333+1+24
8157341*2^33333+1+26
294163565*2^33333+1-28
8612919*2^33333+1+30
67538280*2^33333+1-32
25254168*2^33333+1+34
130627275*2^33333+1+36
6341804*2^33333+1+38
248023491*2^33333+1+40
3481928*2^33333+1+42
69573476*2^33333+1+44
80951568*2^33333+1+46
44273366*2^33333+1+48
28516124*2^33333+1+50
212932466*2^33333+1-52
51507083*2^33333+1+54
64950401*2^33333+1+56
51768264*2^33333+1+58
210464294*2^33333+1+60

131040186*2^33333+1+64
36466745*2^33333+1+66
749474*2^33333+1+68
93371154*2^33333+1+70
120414623*2^33333+1+72
149627024*2^33333+1+74
5933916*2^33333+1+76
11051406*2^33333+1+78
52328084*2^33333+1+80
10867035*2^33333+1+82
129523146*2^33333+1+84
121984445*2^33333+1+86
90390765*2^33333+1+88
248431334*2^33333+1+90
60327905*2^33333+1+92
112053471*2^33333+1+94
42077730*2^33333+1+96
205553789*2^33333+1+98
187658004*2^33333+1+100

Size 62 was harder to crack but here is a quite large case at a prime size where a solution was unexpected:
2147092^8192+1+62, 51871 digits, +1 prime by Fougeron, Proth.exe

This demonstrates that all even gap sizes up to 100 have at least one gigantic occurrence if a prp is allowed. I have not heard of names or previous searches for sizes 8 to 100.

[Joshua2]

Largest sexy prime has been found and verified by Morain and myself. See: http://primes.utm.edu/primes/page.php?id=87280 One of the largest ECPP ever!