mersenneforum.org - View Single Post - Fermat number and Modulo for searching divisors

R.D. Silverman · 2011-05-30, 21:01

Quote:

Originally Posted by CyD

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

It is trivially known that any divisor p of 2^(2^n) + 1 must equal 1 mod
(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.