<s>[INST] $x^2-y^2=2s$, s cannot be an odd integer How can we prove that if $x^2-y^2=2s$ holds, s cannot be an odd integer. What theorem in number theory should we use?
 [/INST] $x^2-y^2=(x+y)(x-y)$
Let $y=x+a$ then
$(x+y)(x-y)=(2x+a)(2x-a)=4x^2-a^2$
If $4x^2-a^2=2s$ then $a^2$ must be divisible by 2. Since $a$ is an integer $a$ must be divisible by 2. Thus $a=2b$. Thus $2s=4x^2-4b^2=4c$ where $c=x^2-b^2$ is an integer. Thus $s$ is divisible by $2$.
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