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 2011-05-30, 19:04 #1 CyD   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 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