<s>[INST] How to show this is relatively prime? Let $a_1= 2$, and for each
$y > 1$, deﬁne $a_{y+1} = a_y(a_y −1) +1$. 
Prove that for all $x \ne y$, $a_x$ and $a_y$ are coprime.
 [/INST] Hint:
$a_{y+1}=a_y(a_y-1)+1$
$\implies  a_{y+1}-1=a_y(a_y-1) $
So, $ a_y-1=a_{y-1}(a_{y-1}-1)$ and $a_{y+1}-1=a_ya_{y-1}(a_{y-1}-1) $
$ a_{y-1}-1=a_{y-2}(a_{y-2}-1)$ and 
so $a_{y+1}-1=a_ya_{y-1}a_{y-2}(a_{y-2}-1)=(a_1-1)\prod_{y\le x \le1}a_x=\prod_{y\le x \le1}a_x$ as $a_1=2$
As $(a_{y+1}-1,a_{y+1})=1, (\prod_{y\le x \le1}a_x,a_{y+1})=1\implies (a_x,a_{y+1})=1$  for $x\le y$
</s>