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Old 2011-05-30, 19:04   #1
CyD
 
May 2011

2 Posts
Default 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 semi-prim (even if it is not semi-prim).
For example of semi-prim, I use a little number N, let it be equal to 105.
 N = 3*5*7=105
Here, N is not semi-prim because it has 3 divisors.
I choose to considerate N like a semi-prim event if it is not.
 N=D_1*D_2 Let  D_1 and  D_2 be  D_1=3 and  D_2 =35 or  D_1 = 5 and  D_2 = 21 or  D_1=7 and  D_2 = 15

About Fermat numbers :

Let define the 2 divisors of  F_m by  D_{m,1} and  D_{m,2} ,
and  X_m and  T_m by:  D_{m,1} = X_m.2^{m+2} +1 and  D_{m,2} = T_m.2^{m+2} +1

So, we have the following properties (for  i \leq i_{max} :
 2^{2^{n}-i.(m+2)} = - (-X)^i mod D_{m,1}
and in an equivalent way :
 2^{2^{n}-i.(m+2)} = - (-T)^i mod D_{m,2}
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
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