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:

[a]-020809-FERMAT AND MERSENNE NUMBER CONJECTURE-(1).nb


References:

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

Advertisements

Actions

Information




%d bloggers like this: