# Number theory with sequences – Mathematics Stack Exchange

The sequence $$x_0,x_1,x_2,x_3,x_4,…..$$ is defined as $$x_0=2$$,$$x_{k+1}=2cdot x_k^2-1$$,for every $$kgeq0$$.Prove that if an odd prime number $$p$$ divides $$x_n$$,then $$2^{n+3}$$ divides $$p^2-1$$

My idea was to firstly treat simple cases like this:

If $$n=0$$,then $$x_0=2$$ doesn’t have any odd divisors.

If $$n=1$$,then $$x_1=7$$.It has only one odd prime divisor,$$7$$ and $$2^{n+3}=2^4=16$$ divides $$p^2-1=7^2-1=48$$

………

From here, my idea was to aply some sort of induction but induction doesn’t work too well with prime numbers so this is where I got stuck with this idea.

Another idea was to write $$x_n$$ in its polynomial form and making it easier to prove what we want.I think that induction might work here but I stil didn’t mange to do it.Can you help me ,please?