# Opinion on proof!

• February 14th 2010, 02:09 PM
gate13
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
• February 16th 2010, 05:43 AM
Hello gate13
Quote:

Originally Posted by gate13
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 $p|n$.