Fermat number factors

[Citrix]

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

[geoff]

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.

[Citrix]

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.

[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?

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.

[geoff]

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.

[geoff]

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

[Citrix]

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.)

[geoff]

Please reserve 300K-400K for me. I will also extend the sieve up to p=1e12 (currently at p=400e9).

[Citrix]

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.

[geoff]

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.

[geoff]

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.