Question 129
Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and
$$
a_{n+1} = a_n^3 - 4c \times a_n^2 + 5c^2 \times a_n + c
$$
for all $n \ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1, \dots, a_{n-1}$.
$$
a_{n+1} = a_n^3 - 4c \times a_n^2 + 5c^2 \times a_n + c
$$
for all $n \ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1, \dots, a_{n-1}$.