FERMAT AND MERSENNE NUMBERS CONJECTURE-(2)

14 02 2009


(2)-CONSTRUCTING SOME FUNCTION ZEROS

f(n)=Mod( \phi(n),\sigma_0(n))

(2.1) PRIMES:

\displaystyle f(p)=Mod(p-1,2)=0, holds for every odd prime p \in \mathbb{P} -\{2\}.

f(2)=1

(2.2) PRODUCT OF DISTINCT PRIMES NOT 2:

\displaystyle f(p_1*p_2*...*p_n)=0, because \displaystyle \sigma_0(p_i)|\phi(p_i) \rightarrow {2 |(p_i-1)}, always holds if \displaystyle p_{i} \neq 2

With \displaystyle k, distinct primes, none of them equal two, it is possible to combine them in \displaystyle 2^n, products to find \displaystyle 2^n zeros.

This set of zeros can be described as odd squarefree numbers [1].

(2.3) POWERS OF 2:

\displaystyle f(2^k)=0\;\rightarrow (k+1)|2^{k-1}, so \displaystyle k+1=2^n must be a power of 2:

\displaystyle k=2^n-1=M_n
A power of 2, is a function zero, iff the exponent is a Mersenne number.

\displaystyle f(2^{M_n})=f(2^{2^{n}-1})=0

(2.4) POWERS OF A PRIME:

\displaystyle f(p^k)=0\;\rightarrow (k+1)|(p-1)*p^{k-1}, then the zeros can fall into two cases:

(2.4.1) Case: \displaystyle (k+1)|(p-1)

Then if \displaystyle d|(p-1), \displaystyle f(p^{d-1})=0, and we can built \displaystyle \sigma_{0}(p-1), zeros, one for every divisor of \displaystyle (p-1).

Note that, in the particular case:

\displaystyle (k+1)=(p-1) \rightarrow{k=p-2}\rightarrow{f(p^{p-2})=0}.

And \displaystyle f(p^{p-1})=Mod((p-1)*p^{p-2},p)=0.

(2.4.2) Case: \displaystyle (k+1)|p^{k-1}

This is more general than (2.3):

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


Archives:

[a]-021409-FERMAT AND MERSENNE NUMBER CONJECTURE-(2).nb


References:

[1]-CRCGreathouse at My Math Forum/Number Theory: Mersenne and Fermat Numbers congruence
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