## FERMAT AND MERSENNE NUMBERS CONJECTURE-(1)

8 02 2009
(1)-INITIAL IDEAS:

Playing, like always, with my computer, I´ve been plotting this function, that includes the Euler totient function, and the Divisor function.

$\displaystyle f(n)=Mod( \phi(n),\sigma_0(n))\: , \; n\in\mathbb{N_{*}^{+}}$

The first 25 values of $\displaystyle f(n)$, (not in OEIS) are:

$\displaystyle \{0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 2,...\}$

Applying $\displaystyle f(x)$ to the Fermat Numbers, $\displaystyle F_n=2^{2^{n}}+1$, and to the Mersenne Numbers, $\displaystyle M_n=2^n-1$, we can conjecture the following congruences:

(1) $\displaystyle \phi(F_n) \equiv 0\; (mod \; \sigma_0(F_n))$

(2) $\displaystyle \phi(F_n-2) \equiv 0\; (mod \; \sigma_0(F_n-2))$

(3) $\displaystyle \phi(M_n) \equiv 0\; (mod \; \sigma_0(M_n))$

(4) $\displaystyle \phi(M_n+2) \equiv 0\; (mod \; \sigma_0(M_n+2))$

¿How do they can be proved? [1]

Archives:

References:

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