Is $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ a square number for integer $x>1$

Question

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.

Answer 1

• This is Problem 12019 in the American Math Monthly. A solution can be found in the link: AMM 12019