Question

if n is a perfect square then 2 n can never be a perfect square. but what if n is equal to 2

Answer 1

Answer:

Let's assume w.o.l.g that a,b>=0

b2=n+2

a2=n

So b2−a2=2, which means (b−a)(b+a)=2. So (b+a) is a divisor of 2 (either 1 or 2, since b+a>0).

Now only a few cases remain

a=0,b=1

a=1,b=0

a=0,b=2

a=1,b=1

a=2,b=0

No case has b2−a2=2

Step-by-step explanation:

If n is a perfect square, then n+2 is not a perfect square.

I also need to state this in first order logic with arithmetic, but have no idea what that looks like.

The only start I have so far in terms of the proof is:

n = a2

n+2 = b2

But I don't know how to proceed from here? I've seen solutions to this already but do not understand how they actually prove anything