20110530, 19:04  #1 
May 2011
2 Posts 
Fermat number and Modulo for searching divisors
Hello,
I try to find somebody who will be able to answer me about the following: I hope it is not too much trouble. May be this property can be used for searching Fermat numbers divisors. I know this forum is not for Fermat numbers, but may be, somebody is able to answer. If you know a forum like this one where you think somebody is able to answer, please, let me know. I demonstrate the following property (All numbers are natural numbers) For a composite Fermat number , I suppose it is semiprim (even if it is not semiprim). For example of semiprim, I use a little number N, let it be equal to 105. Here, N is not semiprim because it has 3 divisors. I choose to considerate N like a semiprim event if it is not. Let and be and or and or and About Fermat numbers : Let define the 2 divisors of by and , and and by: and So, we have the following properties (for : and in an equivalent way : I try to find on the Internet some information about this property but I find nothing. Do you know some internet sites or books about this property ? Do you think this property can be used for searching Fermat numbers divisors? If I'm not clear, please, let me know. Many thanks by advance, Best Regards, Cyril Delestre 
20110530, 21:01  #2  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·313 Posts 
Quote:
(2^(n+2)). I have given proofs on previous occasions. The proof might be given as a homework problem in a first year number theory class. This property is useful for trial division. It is often used to find small divisors for large n. It isn't useful for much of anything else. Last fiddled with by R.D. Silverman on 20110530 at 21:01 Reason: typo 

20110531, 08:16  #3  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
3^{2}·1,303 Posts 
Quote:
Pollard's rho isn't really of much use these days now that ECM is available. Paul 

20110531, 10:52  #4 
May 2011
2 Posts 
I didn't try to prove that any divisor of is like . I know it's known.
I used it in order to demonstrate the following (with the same notation than my previous message) and for example, if and with then and if you have already prove it and if you know some internet site or book, I am interested by that. Cyril 
20110531, 11:24  #5  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×3×313 Posts 
Quote:
Your notation sucks. I can't be bothered wading through it. If you clean it up and repost your comments, I will take a look. Note, however, that trivially m+2 itself is a power of 2. 

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