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

[/code]

I will reserve 994218225 to start with.:popcorn:

Any idea how it compares in terms of computation effort? Things like
how many twin-prime candidates do you end up with for n<250k, the
speed of fixed-k sieving vs. fixed-n sieving. I think fixed-n is faster
but how much?

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.

The fixed n still gives us the best chance.

I took the next k (334639305), and got these primes:

334639305*2^118-1
334639305*2^121-1
334639305*2^127-1
334639305*2^134-1
334639305*2^147-1
334639305*2^147+1
- Twin -
334639305*2^150-1
334639305*2^209-1
334639305*2^233-1
334639305*2^326-1
334639305*2^431-1
334639305*2^478-1
334639305*2^590-1
334639305*2^670-1
334639305*2^751-1
334639305*2^833-1
334639305*2^855-1
334639305*2^866-1
334639305*2^916-1
334639305*2^1014-1
334639305*2^1129-1
334639305*2^1196-1
334639305*2^1360-1
334639305*2^1421-1
334639305*2^1642-1
334639305*2^1756-1
334639305*2^1918-1
334639305*2^1920-1
334639305*2^2210-1
334639305*2^2440-1
334639305*2^2540-1
334639305*2^2605-1
334639305*2^3189-1
334639305*2^3330-1
334639305*2^3508-1
334639305*2^3922-1
334639305*2^4283-1
334639305*2^4601-1
334639305*2^4732-1
334639305*2^5493-1
334639305*2^5854-1
334639305*2^6908-1
334639305*2^8280-1
334639305*2^8794-1
334639305*2^10400-1
334639305*2^10669-1

As you can see, for n=100-11000, there is only one twin. For n=195000, finding a prime is 20 times harder than at n=10000.

[QUOTE=Citrix]There are 11,000 candidates left after sieving for my k. I have found ~10 twin primes already. I am at n=10,000.[/QUOTE]
Unfortunately, none of them are above n=200:


994218225*2^89-1
994218225*2^89+1
- Twin -
994218225*2^108-1
994218225*2^122-1
994218225*2^123-1
- Sophie Germain -
994218225*2^123-1
994218225*2^122-1
- Sophie Germain -
994218225*2^143-1
994218225*2^158-1
994218225*2^208-1
994218225*2^216-1
994218225*2^219-1
994218225*2^237-1
994218225*2^262-1
994218225*2^286-1
994218225*2^293-1
994218225*2^300-1
994218225*2^303-1
994218225*2^312-1
994218225*2^465-1
994218225*2^599-1
994218225*2^843-1
994218225*2^849-1
994218225*2^1026-1
994218225*2^1117-1
994218225*2^1122-1
994218225*2^1165-1
994218225*2^1234-1
994218225*2^1274-1
994218225*2^1385-1
994218225*2^1587-1
994218225*2^1821-1
994218225*2^1831-1
994218225*2^2090-1
994218225*2^2338-1
994218225*2^2675-1
994218225*2^2922-1
994218225*2^3239-1
994218225*2^3494-1
994218225*2^3593-1
994218225*2^4526-1
994218225*2^4708-1
994218225*2^5468-1
994218225*2^6339-1
994218225*2^6805-1
994218225*2^7502-1
994218225*2^9832-1

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.

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)

8 more if anyone wants to attempt.

[code]
451035585 3392
56921865 3331
292777485 3320
261125865 3214
340254915 3200
656771115 3172
118693575 3160
27312285 3089
[/code]

[QUOTE=biwema]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.
[/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 20-30 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.
:coffee: