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

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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.




number-theory elementary-number-theory square-numbers perfect-numbers




share|cite|improve this question

























    up vote
    9
    down vote

    favorite
    2












    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.




    number-theory elementary-number-theory square-numbers perfect-numbers




    share|cite|improve this question























      up vote
      9
      down vote

      favorite
      2









      up vote
      9
      down vote

      favorite
      2






      2





      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.




      number-theory elementary-number-theory square-numbers perfect-numbers




      share|cite|improve this 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.





      number-theory elementary-number-theory square-numbers perfect-numbers





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 24 at 16:50
























      asked Jul 24 at 8:29









      user0410

      1197




      1197




















          1 Answer
          1






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          up vote
          10
          down vote



          accepted










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





          share|cite|improve this answer























          • Is it possible to ask you how to search to find this pdf files? Thanks
            – user0410
            Jul 24 at 8:40






          • 1




            I was lucky to have seen this question before in AMM and was quick to show you a reference :)
            – crskhr
            Jul 24 at 8:43











          • I appreciate you for addressing the link.
            – user0410
            Jul 24 at 8:48






          • 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.
            – gimusi
            Jul 24 at 8:53











          • @gimusi I don't see anyone by the name RobertZ in the MSE users list :(
            – crskhr
            Jul 24 at 11:13










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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          10
          down vote



          accepted










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





          share|cite|improve this answer























          • Is it possible to ask you how to search to find this pdf files? Thanks
            – user0410
            Jul 24 at 8:40






          • 1




            I was lucky to have seen this question before in AMM and was quick to show you a reference :)
            – crskhr
            Jul 24 at 8:43











          • I appreciate you for addressing the link.
            – user0410
            Jul 24 at 8:48






          • 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.
            – gimusi
            Jul 24 at 8:53











          • @gimusi I don't see anyone by the name RobertZ in the MSE users list :(
            – crskhr
            Jul 24 at 11:13














          up vote
          10
          down vote



          accepted










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





          share|cite|improve this answer























          • Is it possible to ask you how to search to find this pdf files? Thanks
            – user0410
            Jul 24 at 8:40






          • 1




            I was lucky to have seen this question before in AMM and was quick to show you a reference :)
            – crskhr
            Jul 24 at 8:43











          • I appreciate you for addressing the link.
            – user0410
            Jul 24 at 8:48






          • 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.
            – gimusi
            Jul 24 at 8:53











          • @gimusi I don't see anyone by the name RobertZ in the MSE users list :(
            – crskhr
            Jul 24 at 11:13












          up vote
          10
          down vote



          accepted







          up vote
          10
          down vote



          accepted






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





          share|cite|improve this answer















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






          share|cite|improve this answer















          share|cite|improve this answer



          share|cite|improve this answer








          edited Jul 24 at 10:04









          Jack M

          17k33473




          17k33473











          answered Jul 24 at 8:35









          crskhr

          3,707825




          3,707825











          • Is it possible to ask you how to search to find this pdf files? Thanks
            – user0410
            Jul 24 at 8:40






          • 1




            I was lucky to have seen this question before in AMM and was quick to show you a reference :)
            – crskhr
            Jul 24 at 8:43











          • I appreciate you for addressing the link.
            – user0410
            Jul 24 at 8:48






          • 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.
            – gimusi
            Jul 24 at 8:53











          • @gimusi I don't see anyone by the name RobertZ in the MSE users list :(
            – crskhr
            Jul 24 at 11:13
















          • Is it possible to ask you how to search to find this pdf files? Thanks
            – user0410
            Jul 24 at 8:40






          • 1




            I was lucky to have seen this question before in AMM and was quick to show you a reference :)
            – crskhr
            Jul 24 at 8:43











          • I appreciate you for addressing the link.
            – user0410
            Jul 24 at 8:48






          • 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.
            – gimusi
            Jul 24 at 8:53











          • @gimusi I don't see anyone by the name RobertZ in the MSE users list :(
            – crskhr
            Jul 24 at 11:13















          Is it possible to ask you how to search to find this pdf files? Thanks
          – user0410
          Jul 24 at 8:40




          Is it possible to ask you how to search to find this pdf files? Thanks
          – user0410
          Jul 24 at 8:40




          1




          1




          I was lucky to have seen this question before in AMM and was quick to show you a reference :)
          – crskhr
          Jul 24 at 8:43





          I was lucky to have seen this question before in AMM and was quick to show you a reference :)
          – crskhr
          Jul 24 at 8:43













          I appreciate you for addressing the link.
          – user0410
          Jul 24 at 8:48




          I appreciate you for addressing the link.
          – user0410
          Jul 24 at 8:48




          2




          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.
          – gimusi
          Jul 24 at 8:53





          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.
          – gimusi
          Jul 24 at 8:53













          @gimusi I don't see anyone by the name RobertZ in the MSE users list :(
          – crskhr
          Jul 24 at 11:13




          @gimusi I don't see anyone by the name RobertZ in the MSE users list :(
          – crskhr
          Jul 24 at 11:13












           

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