# Help about properties of Pell numbers

Hi, I need some help about properties of Pell numbers:

$$U_n = 2 U_{n-1} + U_{n-2} , \text{ with: } U_0=0 \text{ and: } U_1=1$$

$$V_n = 2 V_{n-1} + V_{n-2} , \text{ with: } V_0=2 \text{ and: } V_1=2$$

I have a proof for:

$$\frac{V_{\displaystyle 2^{\scriptstyle n}}}{2} \ = \ 1 + 4 \prod_{i=0}^{n-2}V_{\displaystyle 2^{\scriptstyle i}}^{\scriptstyle 2} \ \equiv \ 1 \pmod{2^{\scriptstyle 2n}} \ \text{\ \ \ \ (for } n \geq 2 )$$

But I have no proof for C1:

$$\frac{V_{\displaystyle 2^{\scriptstyle n}+1}}{2} \ \equiv \ 1 \pmod{2^{\scriptstyle n+1}} \ \text{\ \ \ \ (for } n \geq 2 )$$

and C2:

$$p,q \ \text{ odd primes }, \ \ p \mid V_{\displaystyle q} \ \ \Longrightarrow \ \ p \equiv 1 \pmod{2q}$$

and C3:

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

and C4 (a guess):
$$p \ \text{ odd prime }, \ \ p \mid V_{\displaystyle 2^{\scriptstyle i}} \ \text{ and } \ p = 1 + {(2^{\scriptstyle i}\alpha)}^2 \ \ \Longrightarrow \ \ \alpha = \prod_{j=0}^{k} F_j \ \text{ or } \ \alpha = 1$$.

Can you help ?
Thanks,

Tony

## Answers and Replies

Any idea ?

OK, I've got a proof for C1. It was not so difficult.

About C4, there was a mistake:
$$p \ \text{ odd prime }, \ \ p \mid V_{\displaystyle 2^{\scriptstyle n}} \ \text{ and } \ p = 1 + {(2^{\scriptstyle n}\alpha)}^2 \ \ \Longrightarrow \ \ \alpha = \prod_{j} F_j$$
where $$F_j$$ are Fermat prime numbers.

Any idea ?
tony