TITLE: Is $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ a square number for integer $x>1$
QUESTION [10 upvotes]: 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.

REPLY [11 votes]: This is Problem 12019 in the American Math Monthly. A solution can be found in the link: AMM 12019