Prove that there are no positive integer solutions for $a,b,c,d$ if $a^2-b=c^2, b^2-a=d^2$

Question

Prove that there are no positive integer solutions for $a,b,c,d$ if $$ \left\{ \begin{array}{c} a^2-b=c^2 \\ b^2-a=d^2 \end{array} \right. $$

This question, in my opinion, is hard and I have been attempting it a lot. Here are my attempts:

I tried to use modular arithmetics to prove that if one of the equations is a square then the other cannot but I couldn't.

I also tried adding up the equations and to somehow get that $x^2<b^2-a<(x+1)^2$ but failed.

Any help would be appreciated. Thank you anyways.

Answer 1

Hint: suppose $a\geq b>1$. Then $a^2-b>(a-1)^2$.