Question 47
Let $a_{1}, a_{2}, \ldots$ be the integer sequence with $a_{1}=1$ and, for every integer $n \geqslant 1$,
$$
a_{n+1}=a_{n}^{2}+a_{n}+1
$$
Prove that $a_{n}^{2}+1$ divides $a_{n+1}^{2}+1$ for every integer $n \geqslant 1$.
$$
a_{n+1}=a_{n}^{2}+a_{n}+1
$$
Prove that $a_{n}^{2}+1$ divides $a_{n+1}^{2}+1$ for every integer $n \geqslant 1$.