Školjka

%V0 The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: $a_0=2, \quad a_{k+1}=2a_k^2-1 \quad$ for $k \geq 0$. Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. comment Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u Edited by Orl.