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Proof about integers

  1. Jan 28, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that 2n is representable when n is. Is the converse true?
    Representable is when a positive integer can be written
    as the sum of 2 integral squares.
    3. The attempt at a solution
    so n can be written as [itex] x^2+y^2 [/itex]
    x and y are positive integers
    so then [itex] 2n=2(x^2+y^2) [/itex]
    Im not really sure where to go next, maybe i should look a the prime factors.
    Just to make sure that i know what converse is,
    Would the converse be if 2n is representable so is n.
     
  2. jcsd
  3. Jan 28, 2013 #2

    Dick

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    Brahmagupta–Fibonacci identity. Look it up. And yes, that would be the statement of the converse.
     
  4. Jan 29, 2013 #3

    SammyS

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    Try some specific examples, like 25 = 32 + 42

    or 29 = 22 + 52 .

    Yes, that's the converse.
     
  5. Jan 29, 2013 #4
    ok thanks for the responses. Knowing my teacher I would need to prove the
    Brahmagupta–Fibonacci identity, but I guess I could multiply it out and show that
    the left side equaled the right side. I think the converse is true but ill think about how to prove it.
     
  6. Jan 29, 2013 #5

    Dick

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    Yes, you should definitely think much harder about the converse. And sure, it's much easier to prove them than to discover they exist.
     
    Last edited: Jan 29, 2013
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