Is $2^{57} + 1$ is a composite number? [closed]












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Prove or disprove the following: $2^{57} + 1$ is a composite number.






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closed as off-topic by The Chaz 2.0, Randall, max_zorn, Math Lover, Kemono Chen Jan 25 at 3:27


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    @Randall Yes I know, but she can write the context of the question, it is from of a book or a class or Algebra, etc.
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1












$begingroup$


Prove or disprove the following: $2^{57} + 1$ is a composite number.






elementary-number-theory







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closed as off-topic by The Chaz 2.0, Randall, max_zorn, Math Lover, Kemono Chen Jan 25 at 3:27


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – The Chaz 2.0, Randall, max_zorn, Math Lover, Kemono Chen

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    $begingroup$
    See here: math.stackexchange.com/help/how-to-ask
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    – Randall
    Jan 25 at 3:08










  • $begingroup$
    You must be more explanatory, we don't make homeworks
    $endgroup$
    – El borito
    Jan 25 at 3:19










  • $begingroup$
    @Elborito it doesn't matter because people will answer it anyway
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    – Randall
    Jan 25 at 3:23










  • $begingroup$
    @Randall Yes I know, but she can write the context of the question, it is from of a book or a class or Algebra, etc.
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    – El borito
    Jan 25 at 3:33










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    – lab bhattacharjee
    Jan 25 at 3:52














1












1








1





$begingroup$


Prove or disprove the following: $2^{57} + 1$ is a composite number.






elementary-number-theory







share|cite|improve this question











$endgroup$




Prove or disprove the following: $2^{57} + 1$ is a composite number.






elementary-number-theory




elementary-number-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 25 at 5:30









J. W. Tanner

3,3801320




3,3801320










asked Jan 25 at 3:06









Sunita JainSunita Jain

182




182




closed as off-topic by The Chaz 2.0, Randall, max_zorn, Math Lover, Kemono Chen Jan 25 at 3:27


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – The Chaz 2.0, Randall, max_zorn, Math Lover, Kemono Chen

If this question can be reworded to fit the rules in the help center, please edit the question.







closed as off-topic by The Chaz 2.0, Randall, max_zorn, Math Lover, Kemono Chen Jan 25 at 3:27


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – The Chaz 2.0, Randall, max_zorn, Math Lover, Kemono Chen

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    $begingroup$
    See here: math.stackexchange.com/help/how-to-ask
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    – Randall
    Jan 25 at 3:08










  • $begingroup$
    You must be more explanatory, we don't make homeworks
    $endgroup$
    – El borito
    Jan 25 at 3:19










  • $begingroup$
    @Elborito it doesn't matter because people will answer it anyway
    $endgroup$
    – Randall
    Jan 25 at 3:23










  • $begingroup$
    @Randall Yes I know, but she can write the context of the question, it is from of a book or a class or Algebra, etc.
    $endgroup$
    – El borito
    Jan 25 at 3:33










  • $begingroup$
    math.stackexchange.com/questions/641443/…
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    – lab bhattacharjee
    Jan 25 at 3:52














  • 1




    $begingroup$
    See here: math.stackexchange.com/help/how-to-ask
    $endgroup$
    – Randall
    Jan 25 at 3:08










  • $begingroup$
    You must be more explanatory, we don't make homeworks
    $endgroup$
    – El borito
    Jan 25 at 3:19










  • $begingroup$
    @Elborito it doesn't matter because people will answer it anyway
    $endgroup$
    – Randall
    Jan 25 at 3:23










  • $begingroup$
    @Randall Yes I know, but she can write the context of the question, it is from of a book or a class or Algebra, etc.
    $endgroup$
    – El borito
    Jan 25 at 3:33










  • $begingroup$
    math.stackexchange.com/questions/641443/…
    $endgroup$
    – lab bhattacharjee
    Jan 25 at 3:52








1




1




$begingroup$
See here: math.stackexchange.com/help/how-to-ask
$endgroup$
– Randall
Jan 25 at 3:08




$begingroup$
See here: math.stackexchange.com/help/how-to-ask
$endgroup$
– Randall
Jan 25 at 3:08












$begingroup$
You must be more explanatory, we don't make homeworks
$endgroup$
– El borito
Jan 25 at 3:19




$begingroup$
You must be more explanatory, we don't make homeworks
$endgroup$
– El borito
Jan 25 at 3:19












$begingroup$
@Elborito it doesn't matter because people will answer it anyway
$endgroup$
– Randall
Jan 25 at 3:23




$begingroup$
@Elborito it doesn't matter because people will answer it anyway
$endgroup$
– Randall
Jan 25 at 3:23












$begingroup$
@Randall Yes I know, but she can write the context of the question, it is from of a book or a class or Algebra, etc.
$endgroup$
– El borito
Jan 25 at 3:33




$begingroup$
@Randall Yes I know, but she can write the context of the question, it is from of a book or a class or Algebra, etc.
$endgroup$
– El borito
Jan 25 at 3:33












$begingroup$
math.stackexchange.com/questions/641443/…
$endgroup$
– lab bhattacharjee
Jan 25 at 3:52




$begingroup$
math.stackexchange.com/questions/641443/…
$endgroup$
– lab bhattacharjee
Jan 25 at 3:52










2 Answers
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We can factor $$2^{57}+1 = left( 2^{19} right)^3+1 = left( left(2^{19} right)^2 -2^{19} + 1 right)left( 2^{19} + 1 right)$$ where both factors are greater than $1$. Therefore, $2^{57}+1$ is composite.






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    5












    $begingroup$

    Yes: $2^{57}+1equiv(-1)^{57}+1=-1+1=0mod 3$.






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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3












      $begingroup$

      We can factor $$2^{57}+1 = left( 2^{19} right)^3+1 = left( left(2^{19} right)^2 -2^{19} + 1 right)left( 2^{19} + 1 right)$$ where both factors are greater than $1$. Therefore, $2^{57}+1$ is composite.






      share|cite|improve this answer











      $endgroup$


















        3












        $begingroup$

        We can factor $$2^{57}+1 = left( 2^{19} right)^3+1 = left( left(2^{19} right)^2 -2^{19} + 1 right)left( 2^{19} + 1 right)$$ where both factors are greater than $1$. Therefore, $2^{57}+1$ is composite.






        share|cite|improve this answer











        $endgroup$
















          3












          3








          3





          $begingroup$

          We can factor $$2^{57}+1 = left( 2^{19} right)^3+1 = left( left(2^{19} right)^2 -2^{19} + 1 right)left( 2^{19} + 1 right)$$ where both factors are greater than $1$. Therefore, $2^{57}+1$ is composite.






          share|cite|improve this answer











          $endgroup$



          We can factor $$2^{57}+1 = left( 2^{19} right)^3+1 = left( left(2^{19} right)^2 -2^{19} + 1 right)left( 2^{19} + 1 right)$$ where both factors are greater than $1$. Therefore, $2^{57}+1$ is composite.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 25 at 3:58









          Lee David Chung Lin

          4,39341242




          4,39341242










          answered Jan 25 at 3:22









          JimmyK4542JimmyK4542

          41.3k245107




          41.3k245107























              5












              $begingroup$

              Yes: $2^{57}+1equiv(-1)^{57}+1=-1+1=0mod 3$.






              share|cite|improve this answer









              $endgroup$


















                5












                $begingroup$

                Yes: $2^{57}+1equiv(-1)^{57}+1=-1+1=0mod 3$.






                share|cite|improve this answer









                $endgroup$
















                  5












                  5








                  5





                  $begingroup$

                  Yes: $2^{57}+1equiv(-1)^{57}+1=-1+1=0mod 3$.






                  share|cite|improve this answer









                  $endgroup$



                  Yes: $2^{57}+1equiv(-1)^{57}+1=-1+1=0mod 3$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 25 at 3:15









                  sranthropsranthrop

                  7,1061925




                  7,1061925















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