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Homework Help: Abstract prime factorization proof

  1. Sep 16, 2010 #1
    1. The problem statement, all variables and given/known data
    A positive integer a is called a square if a=n^2 for some n in Z. Show that the integer a>1 is a square iff every exponent in its prime factorization is even.



    2. Relevant equations



    3. The attempt at a solution
    Well, I know a=p1^a1p2^a2....pn^a^n is the definition of prime factorization.
    We let p=2n because any even number squared is an even numbers.
    Not sure how to continue.
     
  2. jcsd
  3. Sep 16, 2010 #2
    I keep trying to work on this one on I'm getting nowhere. Am I on the right track or does anyone have a suggestion?
     
  4. Sep 16, 2010 #3
    It's not an even number; the exponents are even.

    So in the prime factorization of a, [itex]{a_1,a_2,...,a_n}[/itex] are even. Now what does that mean about the square root of a?
     
  5. Sep 16, 2010 #4
    @kathrynag,

    When you have an "iff" problem, you really have to solve two problems. Here's a restatement of the problem to make this more clear:
    Let a > 1 be an integer.

    1. Assume that a is a square. Prove that every exponent in its prime factorization is even.

    2. Assume that every exponent in a's prime factorization is even. Prove that a is a square.

    Does this help?
     
  6. Sep 16, 2010 #5
    Yeah, that does help. I fell better about the 2nd part of the proof.
    The first part still worries me.
    Assume n is a square. Then we have a=n^2
    That's about as far as I get with that one.
    I know what I need to prove, but it's getting there.
    n^2=p1^a1p2^a2....pn^an
     
  7. Sep 16, 2010 #6
    It is also even.
     
  8. Sep 17, 2010 #7
    No. Consider 9. Its prime factorization is [itex] 3^2 [/itex]. Therefore it satisfies the rules of your problem. But it's not even, nor is it the square of an even number.
     
  9. Sep 17, 2010 #8
    Hint: Write out the prime factorization for n and then use that to come up with another prime factorization for n^2.
     
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