[Math] Prove: If n is a perfect square, then n+3 is not a perfect square

Question

Use either direct proof, proof by contrapositive, or proof by contradiction.

Using proof by contradiction method

Assume n is a perfect square and n+3 is a perfect square (proof by
contradiction)

There exists integers x and y such that $n = x^2$ and $n+3 = y^2$

Then $x^2 + 3 = y^2$

Then $3 = y^2 – x^2$

Then $3 = (y-x)(y+x)$

Then $y+x = 3$ and $y-x=1$

Then $x = 1, y = 2$

Since $x = 1$, that implies $n = 1$

this is how far I got

Anyone know what I should do now?

Answer 1

$$ 3=y^2-x^2 \ge (x+1)^2-x^2= 2x+1 $$ which is false if $x>1$.