geometry - Why it is impossible for primitive Pythagoras triplets in integers to be all as powerful numbers?

I had seen an elementary proof for Fermat's last theorem at Quora.

I had checked all the steps (around one page only), where I couldn't catch any error, but I was confused about the last step only that includes the main idea that depends on a right angle triangle (in integers), are impossible to be with all sides being as powerful integers, the author claims this is too elementary to prove, but I don't see why this must be true?, there must be a counter example in large numbers (say more than six digits)!
or this might have a simple proof as stated!

So, can we find such a counter example or prove it simply?