(2)-CONSTRUCTING SOME FUNCTION ZEROS
$latex f(n)=Mod( \phi(n),\sigma_0(n))$
(2.1) PRIMES:
$latex \displaystyle f(p)=Mod(p-1,2)=0$, holds for every odd prime $latex p \in \mathbb{P} -\{2\}$.
$latex f(2)=1$
(2.2) PRODUCT OF DISTINCT PRIMES NOT 2:
$latex \displaystyle f(p_1*p_2*...*p_n)=0$, because $latex \displaystyle \sigma_0(p_i)|\phi(p_i) \rightarrow {2 |(p_i-1)}$, always holds if $latex \displaystyle p_{i} \neq 2$
With $latex \displaystyle k$, distinct primes, none of them equal two, it is possible to combine them in $latex \displaystyle 2^n$, products to find $latex \displaystyle 2^n$ zeros.
This set of zeros can be described as odd squarefree numbers [1].
(2.3) POWERS OF 2:
$latex \displaystyle f(2^k)=0\;\rightarrow (k+1)|2^{k-1}$, so $latex \displaystyle k+1=2^n$ must be a power of 2:
$latex \displaystyle k=2^n-1=M_n$
A power of 2, is a function zero, iff the exponent is a Mersenne number.
$latex \displaystyle f(2^{M_n})=f(2^{2^{n}-1})=0$
(2.4) POWERS OF A PRIME:
$latex \displaystyle f(p^k)=0\;\rightarrow (k+1)|(p-1)*p^{k-1}$, then the zeros can fall into two cases:
(2.4.1) Case: $latex \displaystyle (k+1)|(p-1)$
Then if $latex \displaystyle d|(p-1)$, $latex \displaystyle f(p^{d-1})=0$, and we can built $latex \displaystyle \sigma_{0}(p-1)$, zeros, one for every divisor of $latex \displaystyle (p-1)$.
Note that, in the particular case:
$latex \displaystyle (k+1)=(p-1) \rightarrow{k=p-2}\rightarrow{f(p^{p-2})=0}$.
And $latex \displaystyle f(p^{p-1})=Mod((p-1)*p^{p-2},p)=0$.
(2.4.2) Case: $latex \displaystyle (k+1)|p^{k-1}$
This is more general than (2.3):
$latex \displaystyle f(p^{p^{n}-1})=0$
Archives:
[a]- 021409-FERMAT AND MERSENNE NUMBER CONJECTURE-(2).nb
References:
[1]-CRCGreathouse at My Math Forum/Number Theory: Mersenne and Fermat Numbers congruence
