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 2006-06-22, 06:53 #1 Citrix     Jun 2003 22×397 Posts Fermat number factors I am interested in finding fermat number factors that themselves are generalized fermats. The only known example is 169*2^63686+1. (Found by looking at factors on prothsearch.net) These seem to be extremely rare. Is it possible to predict their density? In order to find more of such factors, I have started testing numbers of the form a^2*2^(2*n)+1. I was just wondering if anyone had any tips on how to approach this problem. - One of the problems is that the sieve program does not work on primes of the form 4X+1 only, it tries to test if primes of the form 4x+3 will divide the generalized fermats also. -Secondly, like using Morehead's theorem and similar theorem can some n (exponent) values be removed from the search? I have already figured out algebric factorization for some of the n values. -Any other ways to speed this up? -Anyone interested in helping out? Thank you
2006-08-06, 23:57   #2
geoff

Mar 2003
New Zealand

13×89 Posts

Quote:
 Originally Posted by Citrix -Anyone interested in helping out?
From the discusion in this thread I gather you plan to test sequences a^(2^y)*2^(2^y)+1 for small a and with y as large as possible?

I can help with some PRP testing if you want to post some candidates.

 2006-08-08, 15:24 #3 Citrix     Jun 2003 22×397 Posts I am not at home this week and do not have access to the files. I have only started sieving/PRPing 3^16 and not anything else. If you want you can start on any other k, or we can sort things out, once I get back, early next week.
2006-08-09, 05:33   #4
Citrix

Jun 2003

22×397 Posts

I have managed to get 3^16 file. It is sieved upto 165 G, so safe to PRP till 200,000 after that I or someone else will have to sieve it more. I am almost at n=100K. I will reserve 100k -200k for you, if that is ok?

My plan is that, if I do not find a prime until 200k, I will leave this k.
The primes so far were
43046721*2^176+1 is prime! Time: 66.749 ms.
43046721*2^1792+1 is prime! Time: 26.007 ms.
43046721*2^19936+1 is prime! Time: 2.898 sec.
(Not a good k to find primes?)

As for finding fermat factors, a prime is more likely to be a fermat factor if k is small, hence I am thinking of only test small k's. 3^16 was just for fun, it is unlikely it will reveal a fermat factor. (Since till k=600 is being tested by prothsearch.net, I was thinking of searching all the perfect squares under 1024.--beyond that the probability of finding a fermat factor is too low)

So between the ranges 600 and 1024 there are only 3 sqaures, namely 625, 729, 961. If you wish, you can choose one of these k's to work on.

(I do not have any sieve files, since I haven't started on the above 3)

Thanks.
Attached Files
 t16_b2_k43046721.txt (190.9 KB, 374 views)

2006-08-10, 07:38   #5
geoff

Mar 2003
New Zealand

13×89 Posts

Quote:
 Originally Posted by Citrix I have managed to get 3^16 file. It is sieved upto 165 G, so safe to PRP till 200,000 after that I or someone else will have to sieve it more. I am almost at n=100K. I will reserve 100k -200k for you, if that is ok?
OK, I will PRP test (3^16)*2^n+1 for 100,000 < n < 200,000.

 2006-08-12, 23:35 #6 geoff     Mar 2003 New Zealand 22058 Posts I finished PRP testing 3^16 for 100,000 < n < 200,000: 3^16*2^168480+1 is prime. I also tested 3^32, 5^16, 7^16, 11^16, 13^16 for 0 < n < 50,000, the following are prime: 3^32*2^160+1 3^32*2^800+1 3^32*2^1568+1 3^32*2^2176+1 5^16*2^288+1 5^16*2^1264+1 5^16*2^7296+1 5^16*2^19648+1 11^16*2^32+1 11^16*2^64+1 11^16*2^112+1 11^16*2^1504+1 13^16*2^96+1 13^16*2^544+1 13^16*2^2688+1
 2006-08-14, 23:41 #7 Citrix     Jun 2003 63416 Posts Code: Primes 43046721*2^176+1 is prime! 43046721*2^1792+1 is prime! 43046721*2^19936+1 is prime! 43046721*2^87520+1 is prime! 43046721*2^168480+1 is prime! Ranges 0-100K Citrix 100-200k geoff 200-300K Citrix (At 250k.)
 2006-08-27, 00:04 #8 geoff     Mar 2003 New Zealand 13·89 Posts Please reserve 300K-400K for me. I will also extend the sieve up to p=1e12 (currently at p=400e9).
2006-08-27, 00:45   #9
Citrix

Jun 2003

22·397 Posts

Quote:
 Originally Posted by geoff Please reserve 300K-400K for me. I will also extend the sieve up to p=1e12 (currently at p=400e9).
I hadn't updated the thread, but I have PRPed to 325K. Could I have the new sieved file. I would like to work on 400-500K.

2006-08-27, 01:02   #10
geoff

Mar 2003
New Zealand

13·89 Posts

Quote:
 Originally Posted by Citrix I hadn't updated the thread, but I have PRPed to 325K. Could I have the new sieved file. I would like to work on 400-500K.
OK, so I will PRP 325K-400K. I will PM you with the sieve file tomorrow (it is running on my home machine), and post it here when it is finished to 1e12, probably in a couple of days.

2006-08-31, 01:51   #11
geoff

Mar 2003
New Zealand

13·89 Posts

Attached is the sieve file for (3^16)*2^n+1, sieved to a little over 1e12. I was getting about 12 minutes per factor on a P3/600, so more sieving is worthwhile if you intend to test all of the candidates.
Attached Files
 sieve1165e9.zip (29.3 KB, 217 views)

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