20060610, 02:28  #1 
Jun 2003
1,579 Posts 
Heavy weight K's
Has anyone thought of working on heavy weight k's to yield a twin prime. If one of these k's will yield a twin prime, then the work of this project can be considerably reduced. I am trying to test these k's to 250,000. I thought I would share them with you. If you are interested you can reserve one for yourself.
Code:
k Weight 994218225 3938 334639305 3914 14549535 3685 431636205 3604 732326595 3553 295840545 3550 315239925 3503 609293685 3458 674128455 3408 383137755 3407 
20060610, 03:18  #2 
Nov 2004
California
2^{3}×3×71 Posts 
Any idea how it compares in terms of computation effort? Things like
how many twinprime candidates do you end up with for n<250k, the speed of fixedk sieving vs. fixedn sieving. I think fixedn is faster but how much? 
20060610, 03:21  #3 
Jun 2003
1,579 Posts 
There are 11,000 candidates left after sieving for my k. I have found ~10 twin primes already. I am at n=10,000. I think the number of twin primes will drop as I go to higher n, but I might find one large one, which is what this project is looking for.

20060610, 04:38  #4 
Apprentice Crank
Mar 2006
2·227 Posts 
The fixed n still gives us the best chance.
I took the next k (334639305), and got these primes: 334639305*2^1181 334639305*2^1211 334639305*2^1271 334639305*2^1341 334639305*2^1471 334639305*2^147+1  Twin  334639305*2^1501 334639305*2^2091 334639305*2^2331 334639305*2^3261 334639305*2^4311 334639305*2^4781 334639305*2^5901 334639305*2^6701 334639305*2^7511 334639305*2^8331 334639305*2^8551 334639305*2^8661 334639305*2^9161 334639305*2^10141 334639305*2^11291 334639305*2^11961 334639305*2^13601 334639305*2^14211 334639305*2^16421 334639305*2^17561 334639305*2^19181 334639305*2^19201 334639305*2^22101 334639305*2^24401 334639305*2^25401 334639305*2^26051 334639305*2^31891 334639305*2^33301 334639305*2^35081 334639305*2^39221 334639305*2^42831 334639305*2^46011 334639305*2^47321 334639305*2^54931 334639305*2^58541 334639305*2^69081 334639305*2^82801 334639305*2^87941 334639305*2^104001 334639305*2^106691 As you can see, for n=10011000, there is only one twin. For n=195000, finding a prime is 20 times harder than at n=10000. 
20060610, 04:58  #5  
Apprentice Crank
Mar 2006
2·227 Posts 
Quote:
994218225*2^891 994218225*2^89+1  Twin  994218225*2^1081 994218225*2^1221 994218225*2^1231  Sophie Germain  994218225*2^1231 994218225*2^1221  Sophie Germain  994218225*2^1431 994218225*2^1581 994218225*2^2081 994218225*2^2161 994218225*2^2191 994218225*2^2371 994218225*2^2621 994218225*2^2861 994218225*2^2931 994218225*2^3001 994218225*2^3031 994218225*2^3121 994218225*2^4651 994218225*2^5991 994218225*2^8431 994218225*2^8491 994218225*2^10261 994218225*2^11171 994218225*2^11221 994218225*2^11651 994218225*2^12341 994218225*2^12741 994218225*2^13851 994218225*2^15871 994218225*2^18211 994218225*2^18311 994218225*2^20901 994218225*2^23381 994218225*2^26751 994218225*2^29221 994218225*2^32391 994218225*2^34941 994218225*2^35931 994218225*2^45261 994218225*2^47081 994218225*2^54681 994218225*2^63391 994218225*2^68051 994218225*2^75021 994218225*2^98321 

20060610, 05:35  #6 
Jun 2003
1,579 Posts 
If you used proth.exe, it only checks if the number is a twin prime upto a certain n level. I think that level is n=200.
I would check all your primes to be twin. With that said, I do not want you to abandon the fixed n idea. It won't take alot of resources to get through these 10 candidates. If we even find 1 twin prime that goes into the top 10 list, I think the effort will be worth it. Hoping I will find a twin prime, I am attempting this effort. If other want to join in, the are most welcome to do so. Who knows, if lucky we might find a large twin prime really soon. edit: Top 20 starts at only 20,000 digits, so I am sure I will be able to get a prime in. Last fiddled with by Citrix on 20060610 at 05:36 
20060610, 18:05  #7 
Mar 2004
3·127 Posts 
Good sieving is very important and horizontal sieving takes almost constant (related to the range size). Twins are randomly distributed (after removing small facors), so vertical sieving is not more effective.
The effort you need to find a prime is about proportional to log(n)^4. If the number is twice as large, it needs 16 times more time to find a twin. Using that information shows, that a 20000 digits twin is relatively easy to find. With optimal parameters you might expect 0.25 years (P4 3.4 GHz) 
20060610, 18:22  #8 
Jun 2003
1,579 Posts 
8 more if anyone wants to attempt.
Code:
451035585 3392 56921865 3331 292777485 3320 261125865 3214 340254915 3200 656771115 3172 118693575 3160 27312285 3089 Last fiddled with by Citrix on 20060610 at 18:22 
20060610, 20:38  #9  
Jun 2003
1,579 Posts 
Quote:
You are right. I am done with my k and did not find any top 5000 primes or any twin primes beyond 10,000. I think it is better to stick to the fixed n, since the sieve for fixed n seems to be 2030 times faster. Based on your formula I will have to do test about 10,000 k's of similar weight before I find a 20,000 digit twin prime. 

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