# 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?

• 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? Oct 30, 2013 at 18:06
• Proof : 5+4 = 9. 5-4 = 1. Oct 30, 2013 at 18:08
• Euler is alive, but is working on rather mundane problems these days :-) Oct 30, 2013 at 18:09
• Well, what language are you working in? You may have to express 16 as $SSSSSSSSSSSSSSSS(0)$, for example. Oct 30, 2013 at 18:10
• What is the "it"? The statement, or its proof? Oct 30, 2013 at 18:23

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.

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

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...

• We have to write the original statement in logical notation aswell right? Oct 30, 2013 at 19:12
• 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 Oct 30, 2013 at 19:14