(2)-CONSTRUCTING SOME FUNCTION ZEROS

**(2.1) PRIMES:**

, holds for every odd prime .

**(2.2) PRODUCT OF DISTINCT PRIMES NOT 2:**

, because , always holds if

With , distinct primes, none of them equal two, it is possible to combine them in , products to find zeros.

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

**(2.3) POWERS OF 2:**

, so must be a power of 2:

A power of 2, is a function zero, iff the exponent is a **Mersenne number**.

**(2.4) POWERS OF A PRIME:**

, then the zeros can fall into two cases:

(2.4.1) Case:

Then if , , and we can built , zeros, one for every divisor of .

Note that, in the particular case:

.

And .

(2.4.2) Case:

This is more general than (2.3):

**Archives:**

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

**References:**

**CRCGreathouse**at My Math Forum/Number Theory: Mersenne and Fermat Numbers congruence