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Definition: A positive integer $m$ is said to be a perfect square if there exists an integer $n$ such that $m = n^2$.

Write a detailed structured proof to prove that there exist two distinct positive integers whose sum and difference are both perfect squares.

How would I write this in logical notation?

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    $\begingroup$ Are you asking about how to write the statement in logical notation, or how to write the proof in logical notation? Do you have an idea for an informal proof? $\endgroup$ – Trevor Wilson Oct 30 '13 at 18:06
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    $\begingroup$ Proof : 5+4 = 9. 5-4 = 1. $\endgroup$ – Euler....IS_ALIVE Oct 30 '13 at 18:08
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    $\begingroup$ Euler is alive, but is working on rather mundane problems these days :-) $\endgroup$ – Trevor Wilson Oct 30 '13 at 18:09
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    $\begingroup$ Well, what language are you working in? You may have to express 16 as $SSSSSSSSSSSSSSSS(0)$, for example. $\endgroup$ – Trevor Wilson Oct 30 '13 at 18:10
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    $\begingroup$ What is the "it"? The statement, or its proof? $\endgroup$ – Trevor Wilson Oct 30 '13 at 18:23
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We want to prove :

$$\exists a \in \Bbb Z_+\; \exists b \in \Bbb Z_+\; \exists m \in \Bbb Z_+\;\exists n \in \Bbb Z_+\; \bigl((a-b=m^2) \land (a+b=n^2)\bigr).$$

Since all the quantifiers are existential, one adequate proof is to find $a,b,m,n$ that satisfy the desired equations. If you start with $m,n$ of the same parity, with $m \lt n$ that is not hard.

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$(a,b)=(2n^2+2n+1,2n^2+2n)$ works.

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hey I'm in the same class as you, and stuck on this as well. This is what I have so far. I'm going off the direct proof structure of the existential (chapter 3.7)

This is my general structure:

Let $a = 40$ and $b = 24$.

Then $a \in \Bbb Z_+$ and $b \in \Bbb Z_+$.

Then $$a+b=40+24=64=8^2 \tag a\label a$$ and $$a-b=40-24=16=4^2 \tag b\label b,$$ so $$a \in \Bbb Z_+,\quad b \in \Bbb Z_+,\quad a+b=x^2,\quad a-b=y^2 \tag c\label c$$

I'm not really sure how to format it properly. Also I'm not sure if line \eqref{a} and line \eqref{b} can/are supposed to be combined like what I did on line \eqref{c}. I'm not even sure if line \eqref{c} is correct format wise...

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  • $\begingroup$ We have to write the original statement in logical notation aswell right? $\endgroup$ – user104350 Oct 30 '13 at 19:12
  • $\begingroup$ hey, not entirely sure :(. you have contact info so we can communicate more easily? full disclosure I'm a bit behind in this course so I'm planning on doing this as I get caught up. I'm looking at the past year solutions to see what they mean by detailed structured proof since I don't know if they want it commented as well when they say detailed proof structure they mean just the formal way we've been doing it $\endgroup$ – MarshmellowJello Oct 30 '13 at 19:14

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