The sequence
,
,
is defined as follows:
for
. Prove that if an odd prime
divides
, then
divides
.
comment
Hi guys ,
Here is a nice problem:
Let be given a sequence
such that
and
. Show that if
is an odd prime such that
then we have ![p^2\equiv 1\pmod{2^{n+3}}](/media/m/8/4/7/847e0cb1f573a5a18a6b96f9b59e2bf4.png)
Here are some futher question proposed by me :Prove or disprove that :
1)![gcd(n,a_n)=1](/media/m/4/0/6/4069385d39e069165d031fd02abb871d.png)
2) for every odd prime number
we have
where
where
or ![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
Thanks kiu si u
Edited by Orl.
![a_0](/media/m/6/1/1/61182666f636aa197c5d27a9b3376b53.png)
![a_1](/media/m/6/1/7/6173ac27c63013385bea9def9ff2b61e.png)
![a_2,](/media/m/7/9/3/793eba203312032e6c8f5876ec56bb75.png)
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
![a_0=2, \quad a_{k+1}=2a_k^2-1 \quad](/media/m/5/5/4/554848057e1a7a9094a7c50a0d13caf8.png)
![k \geq 0](/media/m/7/3/a/73a323fe1d8883d41d88552074e84edb.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
![2^{n+3}](/media/m/1/c/5/1c560e4d3526d03ccb850e29f5fd59b9.png)
![p^2-1](/media/m/e/e/1/ee171e03002215f96368b6c6b774c3a4.png)
comment
Hi guys ,
Here is a nice problem:
Let be given a sequence
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
![a_0=2](/media/m/1/f/5/1f57ed4accf3be5673d0a3736cd24188.png)
![a_{n+1}=2a_n^2-1](/media/m/e/f/0/ef024775ab0f4bb819db1f1fefaa89a7.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![p|a_n](/media/m/b/1/0/b10507d40d037aae2968180811f7de76.png)
![p^2\equiv 1\pmod{2^{n+3}}](/media/m/8/4/7/847e0cb1f573a5a18a6b96f9b59e2bf4.png)
Here are some futher question proposed by me :Prove or disprove that :
1)
![gcd(n,a_n)=1](/media/m/4/0/6/4069385d39e069165d031fd02abb871d.png)
2) for every odd prime number
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![a_m\equiv \pm 1\pmod{p}](/media/m/5/9/5/595db5f3479fa09e5f4825ab4d7b513e.png)
![m=\frac{p^2-1}{2^k}](/media/m/b/2/7/b27d72e1e09e953e88e02a10eb804c91.png)
![k=1](/media/m/f/7/1/f71077af98878d94ee3faacc57dd14b5.png)
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
Thanks kiu si u
Edited by Orl.