NL-NWO 1999 Problem 5

Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$ . Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$ . (a) Suppose that $c = 0$ . Show that $d_n = 1,\ \forall n \geq 1$ . (b) Suppose that $c = 1$ . Show that $d_n \in \{1,5\},\ \forall n \geq 1$ . (c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$ .