Motivated by this question.
How to prove that $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ is not a square number for integer $x>1$?
Thanks for any suggestions.
Edition by the notification of @gimusi:
The answer of this post that is mentioned by @crskhr, is provided by the dear user @Robert Z.
Thanks to all users that have been contributed in this post.