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Help about properties of Pell numbers

  1. Apr 27, 2005 #1
    Hi, I need some help about properties of Pell numbers:

    [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
     
  2. jcsd
  3. Apr 30, 2005 #2
    Any idea ?

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

    About C4, there was a mistake:
    [tex]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 [/tex]
    where [tex]F_j[/tex] are Fermat prime numbers.

    Any idea ?
    tony
     
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