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.

elementary-number-theory

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edited Jan 25 at 5:30

J. W. Tanner

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asked Jan 25 at 3:06

Sunita JainSunita Jain

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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

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

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  – Randall
  Jan 25 at 3:08
  

  
  
  
  


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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
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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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  – El borito
  Jan 25 at 3:33
  

  
  
  
  


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  – lab bhattacharjee
  Jan 25 at 3:52
  

  
  
  
  

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1

$begingroup$

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

elementary-number-theory

share|cite|improve this question

edited Jan 25 at 5:30

J. W. Tanner

3,3801320

asked Jan 25 at 3:06

Sunita JainSunita Jain

182

$endgroup$

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
  $endgroup$
  – 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
  

  
  
  
  

add a comment | 

1

1

1

$begingroup$

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

elementary-number-theory

share|cite|improve this question

edited Jan 25 at 5:30

J. W. Tanner

3,3801320

asked Jan 25 at 3:06

Sunita JainSunita Jain

182

$endgroup$

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

elementary-number-theory

elementary-number-theory

share|cite|improve this question

edited Jan 25 at 5:30

J. W. Tanner

3,3801320

asked Jan 25 at 3:06

Sunita JainSunita Jain

182

share|cite|improve this question

edited Jan 25 at 5:30

J. W. Tanner

3,3801320

asked Jan 25 at 3:06

Sunita JainSunita Jain

182

share|cite|improve this question

share|cite|improve this question

edited Jan 25 at 5:30

J. W. Tanner

3,3801320

edited Jan 25 at 5:30

J. W. Tanner

3,3801320

edited Jan 25 at 5:30

J. W. Tanner

3,3801320

3,3801320

asked Jan 25 at 3:06

Sunita JainSunita Jain

182

asked Jan 25 at 3:06

Sunita JainSunita Jain

182

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
  $endgroup$
  – 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
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  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

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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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edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

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5

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

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answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

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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

edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

$endgroup$

add a comment | 

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

edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

$endgroup$

add a comment | 

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

edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

$endgroup$

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

edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

share|cite|improve this answer

share|cite|improve this answer

edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

edited Jan 25 at 3:58

Lee David Chung Lin

4,39341242

4,39341242

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

answered Jan 25 at 3:22

JimmyK4542JimmyK4542

41.3k245107

41.3k245107

add a comment | 

add a comment | 

5

$begingroup$

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

share|cite|improve this answer

answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

$endgroup$

add a comment | 

5

$begingroup$

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

share|cite|improve this answer

answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

$endgroup$

add a comment | 

5

5

5

$begingroup$

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

share|cite|improve this answer

answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

$endgroup$

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

share|cite|improve this answer

answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

share|cite|improve this answer

share|cite|improve this answer

answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

answered Jan 25 at 3:15

sranthropsranthrop

7,1061925

7,1061925

add a comment | 

add a comment | 

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