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 — Jose Arnaldo Bebita Dris, Jan 28 '20 at 00:10

score 11 · Accepted Answer · edited Jul 24 '18 at 10:04

11

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

edited Jul 24 '18 at 10:04

Jack M

answered Jul 24 '18 at 08:35

C.S.

Is it possible to ask you how to search to find this pdf files? Thanks – user0410 Jul 24 '18 at 08:40
1

I was lucky to have seen this question before in AMM and was quick to show you a reference :) – C.S. Jul 24 '18 at 08:43
I appreciate you for addressing the link. – user0410 Jul 24 '18 at 08:48
2

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. – user Jul 24 '18 at 08:53
@gimusi I don't see anyone by the name RobertZ in the MSE users list :( – C.S. Jul 24 '18 at 11:13
@crskhr Here is https://math.stackexchange.com/users/299698/robert-z – user Jul 24 '18 at 11:42
@gimusi Ah, i missed the space :) Thanks a ton! – C.S. Jul 24 '18 at 11:45

1 Answers1