There are two integers whose sum and difference are perfect squares 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?
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.
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...