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Prove this equation doesn't exist [duplicate]

Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$I have the following Diophantine equation $x=y^2-z^2$Where is the Contradiction? Elementary Number Theory Proof : No natural numbers x, y such that $x^2 - y^2 = 2s$ where s odd integer.Prove that for any integer $k>1$ and any positive integer $n$, there exist $n$ consecutive odd integers whose sum is $n^k$Does there exist some $k$ such that $2^n+k$ is never prime?Does there exist an $a$ such that $a^n+1$ is divisible by $n^3$ for infinitely many $n$?Prove that there doesn't exist any integer $x ge 3$ such that $x^2-1$ is prime.Prove that if m and n are any two odd (integers) then mn is also odd.A natural number written as an arithmetic progressionProve that there exist integers such that the congruence does not holdProve the product of an even integer and an odd integer is even

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0

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This question already has an answer here:

• 
  Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$
  
  8 answers
  
  

Let $n$ be an even integer where $n/2$ is odd.
Prove there doesn't exist $2$ integers $a$ and $b$ such that $a^2-b^2= n$.

number-theory elementary-number-theory discrete-mathematics

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edited Mar 12 at 15:35

Rócherz

2,9863821

asked Mar 12 at 15:24

SergioSergio

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marked as duplicate by Dietrich Burde number-theory
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Mar 12 at 15:29

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What have you tried?
  $endgroup$
  – Dietrich Burde
  Mar 12 at 15:25
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Use $a^2 - b^2 = n$ etc. It will make your text more readable: $a^2 - b^2 = n$ ;)
  $endgroup$
  – Antoine
  Mar 12 at 15:26
  

  
  
  
  

add a comment | 

-4

0

$begingroup$

This question already has an answer here:

• 
  Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$
  
  8 answers
  
  

Let $n$ be an even integer where $n/2$ is odd.
Prove there doesn't exist $2$ integers $a$ and $b$ such that $a^2-b^2= n$.

number-theory elementary-number-theory discrete-mathematics

share|cite|improve this question

edited Mar 12 at 15:35

Rócherz

2,9863821

asked Mar 12 at 15:24

SergioSergio

11

$endgroup$

marked as duplicate by Dietrich Burde number-theory
Users with the  number-theory badge can single-handedly close number-theory questions as duplicates and reopen them as needed.

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Mar 12 at 15:29

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What have you tried?
  $endgroup$
  – Dietrich Burde
  Mar 12 at 15:25
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Use $a^2 - b^2 = n$ etc. It will make your text more readable: $a^2 - b^2 = n$ ;)
  $endgroup$
  – Antoine
  Mar 12 at 15:26
  

  
  
  
  

add a comment | 

$begingroup$

This question already has an answer here:

• 
  Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$
  
  8 answers
  
  

Let $n$ be an even integer where $n/2$ is odd.
Prove there doesn't exist $2$ integers $a$ and $b$ such that $a^2-b^2= n$.

number-theory elementary-number-theory discrete-mathematics

share|cite|improve this question

edited Mar 12 at 15:35

Rócherz

2,9863821

asked Mar 12 at 15:24

SergioSergio

11

$endgroup$

share|cite|improve this question

edited Mar 12 at 15:35

Rócherz

2,9863821

asked Mar 12 at 15:24

SergioSergio

11

share|cite|improve this question

edited Mar 12 at 15:35

Rócherz

2,9863821

asked Mar 12 at 15:24

SergioSergio

11

edited Mar 12 at 15:35

Rócherz

2,9863821

edited Mar 12 at 15:35

Rócherz

2,9863821

asked Mar 12 at 15:24

SergioSergio

11

asked Mar 12 at 15:24

SergioSergio

11

2 Answers
2

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oldest

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0

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Hint: $a^2-b^2 = (a+b)(a-b)$. Investigate the parity of $a+b$ and $a-b$

share|cite|improve this answer

answered Mar 12 at 15:26

KlausKlaus

2,56713

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you, I already tried to go from (a+b)(a-b) but didn't find any method to continue.
  $endgroup$
  – Sergio
  Mar 12 at 15:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Like I said, investigate the parity. What if $a+b$ is even, how about $a-b$ and so on...
  $endgroup$
  – Klaus
  Mar 12 at 15:31
  

  
  
  
  

add a comment | 

0

$begingroup$

Hint: $nequiv 2 bmod 4$, but $a^2-b^2$ is not congruent $2$ modulo $4$.

share|cite|improve this answer

answered Mar 12 at 15:28

Dietrich BurdeDietrich Burde

81k648106

$endgroup$

add a comment | 

0

$begingroup$

Hint: $a^2-b^2 = (a+b)(a-b)$. Investigate the parity of $a+b$ and $a-b$

share|cite|improve this answer

answered Mar 12 at 15:26

KlausKlaus

2,56713

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you, I already tried to go from (a+b)(a-b) but didn't find any method to continue.
  $endgroup$
  – Sergio
  Mar 12 at 15:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Like I said, investigate the parity. What if $a+b$ is even, how about $a-b$ and so on...
  $endgroup$
  – Klaus
  Mar 12 at 15:31
  

  
  
  
  

add a comment | 

0

$begingroup$

Hint: $a^2-b^2 = (a+b)(a-b)$. Investigate the parity of $a+b$ and $a-b$

share|cite|improve this answer

answered Mar 12 at 15:26

KlausKlaus

2,56713

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you, I already tried to go from (a+b)(a-b) but didn't find any method to continue.
  $endgroup$
  – Sergio
  Mar 12 at 15:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Like I said, investigate the parity. What if $a+b$ is even, how about $a-b$ and so on...
  $endgroup$
  – Klaus
  Mar 12 at 15:31
  

  
  
  
  

add a comment | 

$begingroup$

Hint: $a^2-b^2 = (a+b)(a-b)$. Investigate the parity of $a+b$ and $a-b$

share|cite|improve this answer

answered Mar 12 at 15:26

KlausKlaus

2,56713

$endgroup$

share|cite|improve this answer

answered Mar 12 at 15:26

KlausKlaus

2,56713

answered Mar 12 at 15:26

KlausKlaus

2,56713

answered Mar 12 at 15:26

KlausKlaus

2,56713

0

$begingroup$

Hint: $nequiv 2 bmod 4$, but $a^2-b^2$ is not congruent $2$ modulo $4$.

share|cite|improve this answer

answered Mar 12 at 15:28

Dietrich BurdeDietrich Burde

81k648106

$endgroup$

add a comment | 

0

$begingroup$

Hint: $nequiv 2 bmod 4$, but $a^2-b^2$ is not congruent $2$ modulo $4$.

share|cite|improve this answer

answered Mar 12 at 15:28

Dietrich BurdeDietrich Burde

81k648106

$endgroup$

add a comment | 

$begingroup$

Hint: $nequiv 2 bmod 4$, but $a^2-b^2$ is not congruent $2$ modulo $4$.

share|cite|improve this answer

answered Mar 12 at 15:28

Dietrich BurdeDietrich Burde

81k648106

$endgroup$

share|cite|improve this answer

answered Mar 12 at 15:28

Dietrich BurdeDietrich Burde

81k648106

answered Mar 12 at 15:28

Dietrich BurdeDietrich Burde

81k648106

answered Mar 12 at 15:28

Dietrich BurdeDietrich Burde

81k648106