- #1

T.Rex

- 62

- 0

[tex] U_n = 2 U_{n-1} + U_{n-2} , \text{ with: } U_0=0 \text{

and: } U_1=1 [/tex]

[tex] V_n = 2 V_{n-1} + V_{n-2} , \text{ with: } V_0=2 \text{

and: } V_1=2 [/tex]

I have a proof for:

[tex]\frac{V_{\displaystyle 2^{\scriptstyle n}}}{2}

\ = \ 1 + 4 \prod_{i=0}^{n-2}V_{\displaystyle 2^{\scriptstyle

i}}^{\scriptstyle 2} \ \equiv \ 1 \pmod{2^{\scriptstyle 2n}} \

\text{\ \ \ \ (for } n \geq 2 )[/tex]

But I have no proof for C1:

[tex]\frac{V_{\displaystyle 2^{\scriptstyle n}+1}}{2}

\ \equiv \ 1 \pmod{2^{\scriptstyle n+1}} \ \text{\ \ \ \ (for

} n \geq 2 )[/tex]

and C2:

[tex]p,q \ \text{ odd primes }, \ \ p \mid V_{\displaystyle q} \ \ \Longrightarrow \ \ p \equiv 1 \pmod{2q}[/tex]

and C3:

[tex]p \ \text{ odd prime }, \ \ p \mid V_{\displaystyle 2^{\scriptstyle i}} \ \ \Longrightarrow \ \ p \equiv 1 \pmod{2^{\scriptstyle i+2}}[/tex]

and C4 (a guess):

[tex]p \ \text{ odd prime }, \ \ p \mid V_{\displaystyle 2^{\scriptstyle i}} \

\text{ and } \ p = 1 + {(2^{\scriptstyle i}\alpha)}^2 \ \

\Longrightarrow \ \ \alpha = \prod_{j=0}^{k} F_j \ \text{ or } \

\alpha = 1 [/tex].

Can you help ?

Thanks,

Tony