Question 26
The sequence $ a_{0}, a_{1}, a_{2}, \ldots $ is defined as follows:
$$ a_{0}=2,\qquad a_{k+1}=2a_{k}^{2}-1\quad\text{for }k\geq0. $$
Prove that if an odd prime p divides $ a_{n} $, then $ 2^{n+3} $ divides $ p^{2} - 1 $.
$$ a_{0}=2,\qquad a_{k+1}=2a_{k}^{2}-1\quad\text{for }k\geq0. $$
Prove that if an odd prime p divides $ a_{n} $, then $ 2^{n+3} $ divides $ p^{2} - 1 $.