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Proof about difference of squares

  1. May 24, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that an integer n can be represented as a difference of 2 squares if it is either
    odd or divisible by 4, otherwise not. The representation is unique if and only if n is a prime number.
    3. The attempt at a solution
    let x and y be integers so then we have [itex] x^2-y^2=n=(x-y)(x+y) [/itex]
    we would look at the case where x and y are even, then we could factor a 2 out of x-y and x+y so it would be divisible by 4. if x or y was odd then n would be odd. and if x and y were both odd we could factor a 2 out of x-y and x+y. But im not sure how to prove the part
    where n is prime, and that would imply x and y are unique. It seems that if n is prime
    then x-y or x+y has to be 1 or else n would be composite.
     
  2. jcsd
  3. May 24, 2013 #2

    haruspex

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    True, but it isn't x and y both being even that really characterises this case.
    No, think that through again.
     
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