Question 31
The integers $a_0, a_1, a_2, a_3, \dots$ are defined as follows:
$$
a_0 = 1, \quad a_1 = 3, \quad \text{and} \quad a_{n+1} = a_n + a_{n-1} \quad \text{for all } n \ge 1.
$$
Find all integers $n \ge 1$ for which $n a_{n+1} + a_n$ and $n a_n + a_{n-1}$ share a common factor greater than 1.
$$
a_0 = 1, \quad a_1 = 3, \quad \text{and} \quad a_{n+1} = a_n + a_{n-1} \quad \text{for all } n \ge 1.
$$
Find all integers $n \ge 1$ for which $n a_{n+1} + a_n$ and $n a_n + a_{n-1}$ share a common factor greater than 1.