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Math Help - Opinion on proof!

  1. #1
    Junior Member
    Joined
    Oct 2009
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    Opinion on proof!

    Good day to all,

    I have been asked the following question:

    Prove that if p is prime and p|n^2 then p|n

    I have attached my proof and was hoping to get an opinion on it, to see if I have understood the material and where I need to tighten up the proof.

    Thanks
    Attached Thumbnails Attached Thumbnails Opinion on proof!-proof1.jpg  
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  2. #2
    MHF Contributor
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    Hello gate13
    Quote Originally Posted by gate13 View Post
    Good day to all,

    I have been asked the following question:

    Prove that if p is prime and p|n^2 then p|n

    I have attached my proof and was hoping to get an opinion on it, to see if I have understood the material and where I need to tighten up the proof.

    Thanks
    I'm no expert on Number Theory, but, since no-one else has offered any comment, your proof looks basically sound to me. I think you could say, after the line, ' p \in prime decomposition of n^2'
    Assume without loss of generality that p_1 = p. Then:
    n^2 = p^{2a_1}\left(\prod_{i=2}^np_i^{2a_i}\right) ...etc
    Also, when you reach the line:
    n = p^{a_1}\left(\prod_{i=2}^np_i^{a_i}\right)
    I think you can say immediately that p|n.

    Grandad
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