Question 161
Let $a$ and $b$ be two given natural numbers and $(x_n)_{n \ge 0}$ the sequence defined by $x_n = n^2 + a n + b$ for any natural number $n$.
a) Prove that the sequence $(x_n)_{n \ge 0}$ contains infinitely many perfect squares if and only if $a^2 = 4b$.
b) Prove that the sequence $(x_n)_{n \ge 0}$ does not contain an infinite arithmetic progression.
a) Prove that the sequence $(x_n)_{n \ge 0}$ contains infinitely many perfect squares if and only if $a^2 = 4b$.
b) Prove that the sequence $(x_n)_{n \ge 0}$ does not contain an infinite arithmetic progression.