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I'm self-studying abstract algebra (using the free book found on this website: http://abstract.pugetsound.edu/), and am having difficulty solving Problem 19 of Chapter 2's end-of-chapter questions. I have quoted it below:

Let $x,y \in \mathbb{N}$ be relatively prime. If $x\cdot y$ is a perfect square, prove that $x$ and $y$ must both be perfect squares.

The answer/hint in the back of the book is "Use the fundamental theorem of arithmetic."

What I've done:

To me, this fact is obvious. However, I don't know how to represent this in mathematical terms/symbols. In words, I would say:

  1. To be a perfect square, the number in question must have a prime factorization consisting only of primes raised to even powers.
  2. Suppose $x$ is not a perfect square (that is, some prime factor (we will call $p$) of $x$ is not raised to an even power).
  3. To make $x\cdot y$ a perfect square, $p$ must also exist as a prime factor of $y$. This is a contradiction, as $x$ and $y$ are coprime. The same reasoning applies if $y$ is assumed as not a perfect square.

My question:

First, I'd like to know if this is a valid proof. (To me, it appears solid, but there may be something I'm overlooking.)

Second, I'd like to know if there is a more "mathematical" way of representing this proof. (Symbols, etc. that I should use.)

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It looks just fine to me, and I can't think of some 'More mathematical" proon than one using the FTA. –  DonAntonio Dec 4 '12 at 14:59
Is there a proof that uses Bezout's identity ? –  Amr Dec 4 '12 at 15:14
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2 Answers

up vote 4 down vote accepted

Your math is fine, although you might want to mention that you are invoking the fundamental theorem of arithmetic when you factor $x$, $y$, and $xy$ (uniquely) into primes.

Here's a general tip on proof-writing: I seldom come across a proof where I think "this would be clearer if the author used more symbols." Making your proof look more "mathy" is not something you need (or want) to strive for. On the other hand, it's important to write proofs in clear, correct English, with proper punctuation. You've done a good job of this.

It's not necessary to enumerate your steps. As you start writing longer proofs, you will find it tedious to do so. As long as your proof is organized neatly into correct sentences (and paragraphs, if the proof is longer), there's no need to number them.

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Your proof of the contrapositive is a good start, but it misses the keyword: the Fundamental Theorem of Arithmetic.

If $x$ is not a square then $x = p_1^{\alpha_1} \dots p_k^{\alpha_k}$, where at least one of $\alpha_1, \dots , \alpha_k$ are odd. Likewise, $y = q_1^{\beta_1} \dots q_m^{\beta_m}$. Note that $p_i \neq q_j$ as $\gcd(x,y) = 1$. Thus, $xy = p_1^{\alpha_1} \dots p_k^{\alpha_k} q_1^{\beta_1} \dots q_m^{\beta_m}$. Since at least one of the $\alpha_i$ were odd, and $p_i \neq q_j$ for all $(i,j)$, it follows that $xy$ is also not a perfect square.

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ont-> not if you have occasion to make another correction. –  hardmath Dec 4 '12 at 15:24
@hardmath: Thank you! –  JavaMan Dec 4 '12 at 16:28
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