FERMAT AND MERSENNE NUMBERS CONJECTURE-(3)

16 02 2009

(2)-CONSTRUCTING SOME FUNCTION ZEROS (ii)

(2.5) POWERS OF A PRIME * ODD SQUAREFREE:

\displaystyle f(N)=f(p_{1}^{k}*M)=f(p_{1}^{k}*p_2*...*p_n)=0\;\rightarrow (k+1)|p_{1}^{k-1}* \phi\left(\frac{N}{p_{1}^{k-1}}\right) , then the zeros can fall into two cases:

(2.5.1) Case: \displaystyle (k+1)| \phi\left(\frac{N}{p_{1}^{k-1}}\right)

Then if \displaystyle d_i| \phi\left(\frac{N}{p_{1}^{k-1}}\right) , \displaystyle f(p_{1}^{d_{i}-1}*M)=0, and we can built \displaystyle \sigma_{0} \left(\frac{N}{p_{1}^{k-1}}\right) , zeros,

one for every divisor of the product applied to every distinct prime factor

 \displaystyle \prod_{i=1}^{i=n}{ (p_{i}-1)} of \displaystyle N

Note that, in the particular case:

 \displaystyle (k+1)=(p_{i}-1) \rightarrow{k=p_{i}-2}\rightarrow{f(p_{1}^{p_{i}-2}*M)=0}.

And \displaystyle f(p_{1}^{p_{1}-1}*M)=Mod(p_{1}^{p_{1}-2}*\phi\left(\frac{N}{p_{1}^{k-1}}\right),p_{1})=0.

(2.5.2) Case: \displaystyle (k+1)|p_{1}^{k-1}

\displaystyle f(p_{1}^{p_{1}^{n}-1})=0


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