<s>[INST] $a-b \nmid a^2+b^2$ if $a>b+2$ and $\gcd(a,b)=1$ Any two co-prime number $a,b$ with $a>b+2$ we have $a^2+b^2$ is not divisible by $a-b$, $a,b \in \mathbb{N}$. 
But how to prove this?
 [/INST] Since $a-b$ divides $a^2-b^2$, if it also divides $a^2+b^2$ then $a-b$ divides $2b^2$.
Now use that $a$ and $b$ are coprime to see that $a-b$ and $b^2$ are coprime. So by Euclides' lemma $a-b$ divides $2$. This contradicts the condition $a-b>2$.
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