Question 109
A sequence $(a_n)$ is defined by $a_1 = 5$, $a_2 = 13$ and
$$
a_{n+2} = 5a_{n+1} - 6a_n, \forall n \ge 2.
$$
a) Prove that $\gcd(a_n, a_{n+1}) = 1$ for all positive integers $n$.
b) Prove that if $p$ is the prime divisor of $a_{2k}$ then $p-1$ is divisible by $2^{k+1}$ for all non-negative integers $k$.
$$
a_{n+2} = 5a_{n+1} - 6a_n, \forall n \ge 2.
$$
a) Prove that $\gcd(a_n, a_{n+1}) = 1$ for all positive integers $n$.
b) Prove that if $p$ is the prime divisor of $a_{2k}$ then $p-1$ is divisible by $2^{k+1}$ for all non-negative integers $k$.