TITLE: Is there $a,b,c,d\in \mathbb N$ so that $a^2+b^2=c^2$, $b^2+c^2=d^2$?
QUESTION [6 upvotes]: Question:
Are there $a,b,c,d \in \mathbb N$ such that $$a^2 + b^2 = c^2 \ \ \text{and} \ \ b^2 + c^2 = d^2$$
I'm a bit lost here.

REPLY [6 votes]: Note that $c^2-b^2=a^2$, $c^2+b^2=d^2$ $\Rightarrow$ $c^4-b^4=(ad)^2$. 
But here (Wiki) and here (MSE) one can read that
equation
$$x^4-y^4=z^2$$
has no (pairwise coprime) solutions $x,y,z\in \mathbb{N}$.