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.

Removed the perfect-numbers tag, as this question has got nothing to do with perfect numbers — ARNIE BEBITA-DRIS, Jan 28, 2020 at 0:10

Jack M · Accepted Answer · 2018-07-24 10:04:15Z

11

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

Jack M

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answered Jul 24, 2018 at 8:35

C.S.

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$\begingroup$ Is it possible to ask you how to search to find this pdf files? Thanks $\endgroup$
– user0410
Jul 24, 2018 at 8:40
1

$\begingroup$ I was lucky to have seen this question before in AMM and was quick to show you a reference :) $\endgroup$
– C.S.
Jul 24, 2018 at 8:43
$\begingroup$ I appreciate you for addressing the link. $\endgroup$
– user0410
Jul 24, 2018 at 8:48
2

$\begingroup$ By the way the solution provided is from the user @RobertZ. Many others solution a study material is available in his website, I highly suggest that as a great reference. $\endgroup$
– user
Jul 24, 2018 at 8:53
$\begingroup$ @gimusi I don't see anyone by the name RobertZ in the MSE users list :( $\endgroup$
– C.S.
Jul 24, 2018 at 11:13

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