20060622, 06:53  #1 
Jun 2003
3^{2}·5^{2}·7 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 
20060806, 23:57  #2  
Mar 2003
New Zealand
13·89 Posts 
Quote:
I can help with some PRP testing if you want to post some candidates. 

20060808, 15:24  #3 
Jun 2003
3^{2}×5^{2}×7 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. 
20060809, 05:33  #4 
Jun 2003
3^{2}·5^{2}·7 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. 
20060810, 07:38  #5  
Mar 2003
New Zealand
13×89 Posts 
Quote:


20060812, 23:35  #6 
Mar 2003
New Zealand
13×89 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 
20060814, 23:41  #7 
Jun 2003
3^{2}·5^{2}·7 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 0100K Citrix 100200k geoff 200300K Citrix (At 250k.) 
20060827, 00:04  #8 
Mar 2003
New Zealand
13·89 Posts 
Please reserve 300K400K for me. I will also extend the sieve up to p=1e12 (currently at p=400e9).

20060827, 00:45  #9 
Jun 2003
3^{2}×5^{2}×7 Posts 

20060827, 01:02  #10 
Mar 2003
New Zealand
13×89 Posts 
OK, so I will PRP 325K400K. 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.

20060831, 01:51  #11 
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.

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